A Novel ξ-Order Hölder Function Classes and Their Integral Transform

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DOI: https://doi.org/10.28924/2291-8639-23-2025-159
Published: Jun 26, 2025
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Xiaohui Gao, A Novel ξ-Order Hölder Function Classes and Their Integral Transform, Int. J. Anal. Appl., 23 (2025), 159.

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Xiaohui Gao

Abstract

We introduce a function class \(\mathcal{H}(\xi)\) (\(\xi > 0\)) capturing tail decay and H\"older regularity. For \(h \in \mathcal{H}(\xi)\), its Fourier transform \(\mathcal{F}[h]\) inherits H\"older continuity of order \(\xi\) and essential boundedness. For \(\xi > 1\), derivatives of \(\mathcal{F}[h]\) up to order \(\lfloor \xi \rfloor\) are \(L^{\infty}\)-bounded, with fractional H\"older continuity arising from \(h\)'s decay. Our approach integrates multiscale analysis and Fourier multiplier theory, extending prior results on H\"older-Fourier correspondences. Novel integral estimates and phase cancellation methods resolve critical gaps in non-integer smoothness characterization. These results deepen the Fourier regularity analysis for non-integer \(\xi\), offering tools for harmonic analysis and pseudo-differential operators.

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