Haar Frame Characterizations of Besov–Sobolev Spaces and Optimal Embeddings into Their Dyadic Counterparts

Type: Article

Publication Date: 2023-06-01

Citations: 1

DOI: https://doi.org/10.1007/s00041-023-10013-7

Abstract

Abstract We study the behavior of Haar coefficients in Besov and Triebel–Lizorkin spaces on $${\mathbb R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness $$s&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , in which the spaces $$F^s_{p,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:math> and $$B^s_{p,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:math> are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that $$1/p&lt;s&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$f\in B^s_{p,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , we actually prove that the usual Haar coefficient norm, $$\Vert \{2^j\langle f, h_{j,\mu }\rangle \}_{j,\mu }\Vert _{b^s_{p,q}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mo>{</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:msup> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:msub> </mml:mrow> </mml:math> remains equivalent to $$\Vert f\Vert _{B^s_{p,q}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>f</mml:mi> <mml:mo>‖</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:msub> </mml:math> , i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case $$s=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$q=\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , we show that such an expression gives an equivalent norm for the Sobolev space $$W^{1}_p({\mathbb R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$1&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , which is related to a classical result by Bočkarev. Finally, in several endpoint cases we give optimal inclusions between $$B^s_{p,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:math> , $$F^s_{p,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:math> , and their dyadic counterparts.

Locations

  • Journal of Fourier Analysis and Applications - View - PDF

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