Carleson measures on spaces of homogeneous type

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a space of homogeneous type in the sense of Coifman and Weiss <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket CW 2 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CW</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[{\text {CW}}2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript plus Baseline equals upper X times bold upper R Superscript plus"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{X^ + } = X \times {{\mathbf {R}}^ + }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A positive function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{X^ + }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to have horizontal bounded ratio <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis HBR right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>HBR</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">({\text {HBR}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{X^ + }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if there is a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma t right-parenthesis less-than-or-equal-to upper A Subscript upper F Baseline upper F left-parenthesis y comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>F</mml:mi> </mml:msub> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F(x,t) \leq {A_F}F(y,t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho left-parenthesis x comma y right-parenthesis greater-than t"> <mml:semantics> <mml:mrow> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho (x,y) &gt; t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (By Harnack’s inequality, a well-known example is any positive harmonic function in the upper half plane.) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="HBR"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>HBR</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {HBR}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a rich class that is closed under a wide variety of operations and it provides useful majorants for many classes of functions that are encountered in harmonic analysis. We are able to prove theorems in spaces of homogeneous type for functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="HBR"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>HBR</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {HBR}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are analogous to the classical Carleson measure theorems and to extend these results to the functions which they majorize. These results may be applied to obtain generalizations of the original Carleson measure theorem, and of results of Flett’s which contain the Hardy-Littlewood theorems on intermediate spaces of analytic functions. Hörmander’s generalization of Carleson’s theorem is included and it is possible to extend those results to the atomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces of Coifman and Weiss.

Locations

• Transactions of the American Mathematical Society - View - PDF