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arXiv · 2504.04137

On a class of Mikhlin multipliers which do not preserve $L^1$-, $L^\infty$-regularity and continuity

Pavel Dimovski·Stevan Pilipovic·Bojan Prangoski
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Abstract

We show that every Fourier multiplier with real-valued and positively homogeneous symbol of order 0, supported in a cone whose dual cone has a nonempty interior and such that the average of the positive part is sufficiently larger than the average of the negative part does not preserve the $L^1$- nor the $L^\infty$ regularity and neither the continuity.We also construct wave front sets which measure the microlocal regularity with respect to a large class of Banach spaces. As a consequence of the first part, we argue that one can never construct wave front sets that behave in a natural way and measure the microlocal $L^1$- nor $L^\infty$-regularity and neither the continuity

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Pavel Dimovski, Stevan Pilipovic, Bojan Prangoski. 2025-04-05. On a class of Mikhlin multipliers which do not preserve $L^1$-, $L^\infty$-regularity and continuity. https://arxiv.org/abs/2504.04137

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