It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem. These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an …
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem. These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers. We also improve a recent extension of Trudingerâs inequality due to Strichartz.
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation …
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers.We also improve a recent extension of Trudinger's inequality due to Strichartz. JjlJl!The following theorem is due to Hardy and Littlewood [3] for d= 1, and to Sobolev [8] in the general case.A simple proof is given in [9, V.l.2].Theorem 1.Let 0<a<d, l<^<^<co, and \lq=\jp-a/d.ThenWÁDhúAWfWr,.Iffis supported by a ball B, and l/9=l-a/d, then Iaif) e L"iB) if$B l/l log+ l/l dx< oo.We first prove a simple lemma.
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The …
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The problem studied in this paper is whether or not rational functions with poles off E are dense in L%(E) (or in L"(E) in the case when E has no interior).For 1 £p S 2 the problem has been settled by Bers and Havin.By a method which applies for 1 sSp<co we give new results for p>2 which improve earlier results by Sinanjan.The results are given in terms of capacities.
Introduction. Notation A class of function spaces Differentiability and spectral synthesis Luzin type theorems Appendix. Whitney's approximation theorem in $L_p(\mathbf{R}^N), p>0$ Bibliography.
Introduction. Notation A class of function spaces Differentiability and spectral synthesis Luzin type theorems Appendix. Whitney's approximation theorem in $L_p(\mathbf{R}^N), p>0$ Bibliography.
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The …
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The problem studied in this paper is whether or not rational functions with poles off E are dense in L%(E) (or in L"(E) in the case when E has no interior).For 1 £p S 2 the problem has been settled by Bers and Havin.By a method which applies for 1 sSp<co we give new results for p>2 which improve earlier results by Sinanjan.The results are given in terms of capacities.
A presentation of some of the highlights in Vladimir Maz'ya's remarkable early work on function spaces, potential theory, and partial differential operators.
A presentation of some of the highlights in Vladimir Maz'ya's remarkable early work on function spaces, potential theory, and partial differential operators.
Introduction. Notation A class of function spaces Differentiability and spectral synthesis Luzin type theorems Appendix. Whitney's approximation theorem in $L_p(\mathbf{R}^N), p>0$ Bibliography.
Introduction. Notation A class of function spaces Differentiability and spectral synthesis Luzin type theorems Appendix. Whitney's approximation theorem in $L_p(\mathbf{R}^N), p>0$ Bibliography.
A presentation of some of the highlights in Vladimir Maz'ya's remarkable early work on function spaces, potential theory, and partial differential operators.
A presentation of some of the highlights in Vladimir Maz'ya's remarkable early work on function spaces, potential theory, and partial differential operators.
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem. These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an …
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem. These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers. We also improve a recent extension of Trudingerâs inequality due to Strichartz.
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation …
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers.We also improve a recent extension of Trudinger's inequality due to Strichartz. JjlJl!The following theorem is due to Hardy and Littlewood [3] for d= 1, and to Sobolev [8] in the general case.A simple proof is given in [9, V.l.2].Theorem 1.Let 0<a<d, l<^<^<co, and \lq=\jp-a/d.ThenWÁDhúAWfWr,.Iffis supported by a ball B, and l/9=l-a/d, then Iaif) e L"iB) if$B l/l log+ l/l dx< oo.We first prove a simple lemma.
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The …
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The problem studied in this paper is whether or not rational functions with poles off E are dense in L%(E) (or in L"(E) in the case when E has no interior).For 1 £p S 2 the problem has been settled by Bers and Havin.By a method which applies for 1 sSp<co we give new results for p>2 which improve earlier results by Sinanjan.The results are given in terms of capacities.
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The …
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The problem studied in this paper is whether or not rational functions with poles off E are dense in L%(E) (or in L"(E) in the case when E has no interior).For 1 £p S 2 the problem has been settled by Bers and Havin.By a method which applies for 1 sSp<co we give new results for p>2 which improve earlier results by Sinanjan.The results are given in terms of capacities.
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The …
Let £ be a compact set in the plane, let L"(E) have its usual meaning, and let Lpa(E) be the subspace of functions analytic in the interior of E. The problem studied in this paper is whether or not rational functions with poles off E are dense in L%(E) (or in L"(E) in the case when E has no interior).For 1 £p S 2 the problem has been settled by Bers and Havin.By a method which applies for 1 sSp<co we give new results for p>2 which improve earlier results by Sinanjan.The results are given in terms of capacities.
operator of order m with infinitely differentiable coefficients defined in an open subset Q of Rn. If w C Q is open, we denote by 9% (w) the space of …
operator of order m with infinitely differentiable coefficients defined in an open subset Q of Rn. If w C Q is open, we denote by 9% (w) the space of distributions u defined in @ which satisfy the homogeneous equation P (x, D) u 0 in f. If F is a relatively closed subset of Q, we denote by 91 (F) the set of all distributions u defined and satisfying the equation P (x, D) u = 0 in a neighborhood of F. For 1 ?p <oo we set Pv(F) ==LP(F) n '(F), the set of functions u E LV(F) which satisfy the equation P (x, D) u -0 in the interior of F. If K C Q is compact then clearly '% (K) C 9VP(K). In this paper we seek conditions on K which will insure that 92 (K) is dense in 'vP(K) with respect to the LP(K) norm. For p oo it is natural to denote by 91 (K) the set of continuous functions on K which satisfy the equation in K. The problem then is to find conditions on K for which ST (K) is dense in 9i (K) in the Liniform norm.
CONTENTS Introduction Chapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures …
CONTENTS Introduction Chapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures § 3. On removable singularities of analytic functions § 4. The analytic C-capacity of sets § 5. Estimates of the coefficients in the Laurent series § 6. The change in the capacity under a conformal transformation of a set Chapter II. The separation of singularities of functions § 1. The construction of a special system of partitions of unity § 2. Integral representations of continuous functions § 3. Separation of singularities § 4. The approximation of functions in parts § 5. Approximation of functions on sets with empty inner boundary § 6. The additivity of capacity for some special partitions of a set Chapter III. Estimation of the Cauchy integral § 1. Statement of the result § 2. Estimate of the Cauchy integral § 3. Estimation of the Cauchy integral along a smooth Lyapunov curve § 4. Proof of the principal theorem § 5. Some consequences § 6. A refinement of the Maximum Principle and the capacity analogue to the theorem on density points Chapter IV. Classification of functions admitting an approximation by rational fractions § 1. Examples of functions that cannot be approximated by rational fractions § 2. A criterion for the approximability of a function § 3. Properties of the second coefficient in the Laurent series § 4. Proof of the principal lemma § 5. Proof of the theorems of § 2 Chapter V. The approximation problem for classes of functions § 1. Removal of the poles of approximating functions from the domain of analyticity of the function being approximated § 2. Necessary conditions for the algebras to coincide § 3. A criterion for the equality of the algebras § 4. Geometrical examples § 5. Some problems in the theory of approximation Chapter VI. The approximation of functions on nowhere dense sets § 1. The instability of capacity § 2. A capacity criterion for the approximability of functions on nowhere dense sets § 3. Theorems on the approximation of continuous functions in terms of Banach algebras § 4. A capacity characterization of the Mergelyan function and of peak points References
CONTENTSIntroduction Part I § 1. The space § 2. -potentials of generalized functions with finite -energy § 3. The maximum principle. A generalization of a theorem of Evans-Vasilesko. Lemmas on …
CONTENTSIntroduction Part I § 1. The space § 2. -potentials of generalized functions with finite -energy § 3. The maximum principle. A generalization of a theorem of Evans-Vasilesko. Lemmas on sequences of potentials § 4. The -capacity of a compact set. The capacity potential § 5. The capacity potential of an analytic set. The measurability of an analytic set relative to -capacity and -complete functions § 6. An estimate of the potential in terms of the modulus of continuity of the measure § 7. Metric properties of sets of zero -capacity § 8. Use of Bessel potentials (the case ). The capacity § 9. Guide to the literature Part II § 1. Auxiliary information on the spaces § 2. Generalized functions with finite -energy and -potential § 3. The maximum principle. A generalization of a theorem of Evans-Vasilesko § 4. The -capacity of a compact set. The capacity potential § 5. The capacity potential of an analytic set. Measurability of analytic sets relative to -capacity and -complete functions § 6. An estimate of the potential in terms of the modulus of continuity of the measure § 7. Metric properties of sets of zero -capacity § 8. Use of Bessel potentials (the case ). The capacity References
Soit Ω un ouvert quelconque connexe de R n . Soit E un espace vectoriel de distributions sur Ω, séparé et complet. On désigne par BL m (E) l’espace des …
Soit Ω un ouvert quelconque connexe de R n . Soit E un espace vectoriel de distributions sur Ω, séparé et complet. On désigne par BL m (E) l’espace des distributions sur Ω dont toutes les dérivées d’ordre m sont dans E. Ces espaces sont les espaces du type de Beppo Levi. Si E=L 2 (Ω), on écrit BL=BL(Ω) au lieu de BL 1 (L 2 (Ω)). La première partie est consacrée aux propriétés générales des espaces BL 1 (E) ; la seconde associe à toute fonction F∈BL(Ω) une fonction “précisée”, définie partout sauf sur un ensemble de capacité extérieure nulle ; la troisième aborde l’étude des BL m (E). Parmi les résultats principaux, signalons : 1) l’espace séparé associé à BL m (E) est complet ; 2) si n>2, l’espace D ^ 1 (Ω), complété de D(Ω) pour la norme ∥ϕ∥ 1 =(∫ Ω | grad → ϕ| 2 dx) 1/2 est un espace de distributions. Si n=2, il en est encore ainsi, si et seulement si, Ω est de complémentaire non polaire ; 3) pour que le problème de Neumann relatif à un ouvert connexe borné Ω et à l’équation Δu=0 soit possible (en un certain sens) il faut et il suffit que Ω soit un ouvert de Nikodym (i.e. toute F∈BL(Ω) est dans L 2 (Ω)) ; 4) pour qu’une F∈BL(Ω) soit dans D ^ 1 (Ω), il faut et il suffit qu’une fonction “précisée” F * , associée à F, admette la pseudo-limite 0 quasi partout à la frontière, et à l’infini si n>2 et Ω non borné, etc.
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation …
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem.These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers.We also improve a recent extension of Trudinger's inequality due to Strichartz. JjlJl!The following theorem is due to Hardy and Littlewood [3] for d= 1, and to Sobolev [8] in the general case.A simple proof is given in [9, V.l.2].Theorem 1.Let 0<a<d, l<^<^<co, and \lq=\jp-a/d.ThenWÁDhúAWfWr,.Iffis supported by a ball B, and l/9=l-a/d, then Iaif) e L"iB) if$B l/l log+ l/l dx< oo.We first prove a simple lemma.
We intend to investigate the connection between Ms(E) and the potential theoretic ~-capacity of E. As Ms(E)= Ms(E), where E is the closure of E, the only case of interest …
We intend to investigate the connection between Ms(E) and the potential theoretic ~-capacity of E. As Ms(E)= Ms(E), where E is the closure of E, the only case of interest is to consider compact sets.The investigation has a close connection with [7], to which we shall refer concerning some details of the proofs.Let us first introduce some notations.The support of a measure # and of a function / is denoted by S~ and S r respectively.S(r), r > 0, is the closed sphere J x I ~< r.The ~-potential, 0 ~< ~< m, of a measure ~u is denoted by u~, where u"~(x)-(d~(y)-31~., if O<~<m,Here and elsewhere, the integration is to be extended over the whole space, if no limits of integration are indicated.If # is absolutely continuous and has a density /, dlu =/dx, we also write u~ instead of u~.If I~(~u) denotes the energy integral of lu, L(~) = f u"~ d~(x),
The abbreviations M~, 21f,~(Zo) or M~($2) will be used when no misunderstanding can result.It will be assumed that,(s2) is not empty.The class ~ is said to be mo~wtonic if s …
The abbreviations M~, 21f,~(Zo) or M~($2) will be used when no misunderstanding can result.It will be assumed that,(s2) is not empty.The class ~ is said to be mo~wtonic if s s implies ~($2)< ~($2').By (I) we have then(2) (s0, $2) =< (s0, $23.Suppose now that z'= h(z) defines a one to one conformal mapping of $2 onto a region $2', and set Zo ~ h(z0).We shall say that the class ~ is con formally) for all such mappings.For a conformally invariant class we have evidentlyThis can be written in the more symmetric formand it is seen that the differential(5) M,~ (~, $2) I ~.Zz I defines a conformally invariant metric in $2. M~ is itself a relative conformal invariant, and this is the type of invariant we shall be mainly concerned with.Absolute invariants can be introduced either as ~he quotient of two relative invariants or by forming the curvature of the metric (5).If ~ is both monotonic and conformally invariant we can combine (2) and (4) to obtain (6) M (eo, $2')'ld gl =< $2)" Ide0l whenever z'= h(z) maps $2 conformally and one to one onto a subregion of $2'.We shall refer to (6) as the weak monotonic property of M~.A ~tronger result is obtained if ~ is analytically invariant.By this we mean that f(z') E ~ (s implies f(h (z)) E ~ ($2) whenever h (z') is single-valued and analytic in $2 with values in $2', regardless of whether h(z) is univalent or not.Since analytic invariance implies conformal invariance the metric (5) will have the same invariance property as before.An analytically invariant class is eo ipso monotonic.Hence (6) is valid, but the stronger assumption implies that (6) holds not only for one to one mappings, but for arbitrary analytie mappings of $2 into $2'.In this ease we shall say that M,~ has the strong monotonic property.
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem. These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an …
It is proved that certain convolution inequalities are easy consequences of the Hardy-Littlewood-Wiener maximal theorem. These inequalities include the Hardy-Littlewood-Sobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers. We also improve a recent extension of Trudingerâs inequality due to Strichartz.
This paper is a direct outgrowth of a conversation the author had with Professor V. G. Maz'ya in the Spring of 1974 concerning the existence of certain Lp estimates for …
This paper is a direct outgrowth of a conversation the author had with Professor V. G. Maz'ya in the Spring of 1974 concerning the existence of certain Lp estimates for the restriction of Riesz potentials of Lp functions to small sets. The main open question concerned the validity of the estimate llI,,f[]v,~ 2 (re<n, l < p < n / m ) given that the measure p satisfies I~(K)~C.R, , ,p (K ) for all compact sets K in R. (See section 1 for the definitions.) Maz'ya has shown ([11] and [12]) that this is valid for m = 1, 2. We now establish this estimate for the remaining integer values of m Theorem 4 of section 4. The proof relies on the following theorem, which is the principal estimate of this paper.
It is shown that two definitions for an L" capacity (1 <p<co) on subsets of Euclidean R" are equivalent in the sense that as set functions their ratio is bounded …
It is shown that two definitions for an L" capacity (1 <p<co) on subsets of Euclidean R" are equivalent in the sense that as set functions their ratio is bounded above and below by positive finite constants.The classical notions of capacity correspond to the case/>=2.1.Let Lx,P=gx(Lv), l<p<oo, a>0, where gx is the Lx function on Rn which is the Fourier transform of (277)-n/2(l + |f| Ta/2, £eRn.L"= LP(Rn) are the usual Lebesgue spaces.Definition 1. £a.j,(^)=inf ||/||£, where the infimum is over allfe L% for which gx *f(x)=l on A, A^Rn.Definition 2. Cx,v(K)=inf \\<p\\XêV, where the infimum is over all cp e C0x(Rn) for which cp(x) = 1 in a neighborhood of K, AT compact set in Rn.Here \\-\\x,v denotes the usual norm in L^and [|-||0"=||-||".The purpose of this paper is to show Theorem A. For all compact sets K<^Rn, BX.P(K)~CX.P(K).Here ~ means that the ratio is bounded above and below by positive finite constants independent of the set K.Remark.The set function CX,P is extended to the class of all subsets by CXmP(A)=sup CX,P(K), where the supremum is over all compact sets K contained in A. When this is done, the equivalence of the theorem extends to all capacitable sets and in particular to all analytic sets.For details see [13].As an example of the utility of these capacities, we state the following removable singularity theorem.Let P(x, D) he a partial differential operator of order m defined in an open set Q^Rn.
Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and …
Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.
Let f (x) and K (x) be two functions integrable over the interval (-∞,+∞). It is very well known that their composition $$ \int\limits_{{ - \infty }}^{{ + \infty }} …
Let f (x) and K (x) be two functions integrable over the interval (-∞,+∞). It is very well known that their composition
$$ \int\limits_{{ - \infty }}^{{ + \infty }} {f(t)K\left( {x - t} \right)dt} $$
exists, as an absolutely convergent integral, for almost every x. The integral can, however, exist almost everywhere even if K is not absolutely integrable. The mostinteresting special case is that of K (x) = 1/x. Let us set
$$ \tilde{f}(x) = \frac{1}{\pi }\int\limits_{{ - \infty }}^{{ + \infty }} {\frac{{f(t)}}{{x - t}}dt} $$
.
A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.
A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.
Conditions are obtained under which the Dirichlet problem for elliptic equations of order , ( is the dimensionality of the space), is stable.Bibliography: 10 titles.
Conditions are obtained under which the Dirichlet problem for elliptic equations of order , ( is the dimensionality of the space), is stable.Bibliography: 10 titles.
In the inner integral we make the substitution x -t = (s-t)u and find that the integral does not exceed | î -/ Ia"1 Í* | u(l -u) \"l2-Hu = …
In the inner integral we make the substitution x -t = (s-t)u and find that the integral does not exceed | î -/ Ia"1 Í* | u(l -u) \"l2-Hu = B(a) | 5 -/I"-1. J -00Consequently, the right-hand side of (3) is dominated by
It is shown that many maximal functions defined on the ${L_p}$ spaces are bounded operators on ${L_p}$ if and only if they satisfy a capacitary weak type inequality.
It is shown that many maximal functions defined on the ${L_p}$ spaces are bounded operators on ${L_p}$ if and only if they satisfy a capacitary weak type inequality.
Necessary and sufficient conditions are found for continuity, compactness, and closability of imbedding operators of some function spaces into the space Lp. These results (for p = 2) give criteria …
Necessary and sufficient conditions are found for continuity, compactness, and closability of imbedding operators of some function spaces into the space Lp. These results (for p = 2) give criteria for positive definiteness and discreteness of the spectrum of the Dirichlet problem for a selfadjoint elliptic operator of arbitrary order. Some integral inequalities are considered for differentiable functions on a cube.
In this paper the authors define a capacity for a given linear partial differential operator acting on a Banach space of distributions. This notion has as special cases Newtonian capacity, …
In this paper the authors define a capacity for a given linear partial differential operator acting on a Banach space of distributions. This notion has as special cases Newtonian capacity, analytic capacity, and <italic>AC</italic> capacity. It is shown that the sets of capacity zero are precisely those sets which are removable sets for the corresponding homogeneous equation. Simple properties of the capacity are derived and special cases examined.
We characterize the compact sets K in the n-dimensional Euclidean space with capacity zero relative to a certain kernel as exactly those sets for which every continous function on K …
We characterize the compact sets K in the n-dimensional Euclidean space with capacity zero relative to a certain kernel as exactly those sets for which every continous function on K has an extensio ...