A measure is called $L^p$-improving if it acts by convolution as a bounded operator from $L^q$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^p$-improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^p$-improving.

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to $l^2$.We also give examples of non-symmetry...

MSC 2010: 35R11, 42A38, 26A33, 33E12The method of integral transforms based on joint application of a fractional generalization of the Fourier transform and the classical Laplace transform is utilized for solving Cauchy-type problems for the time-space fractional diffusion-wave equations expressed in terms of the Caputo time-fractional derivative and the Weyl space-fractional operator. The solutions obtained are in integral form whose kernels are Green functions expressed in terms of the Fox H-functions....

We consider the Fourier transform in the space of Henstock-Kurzweil integrable functions. We prove that the classical results related to the Riemann-Lebesgue lemma, existence and continuity are true in appropriate subspaces.

Writing $(T_R^{\lambda }f)\widehat{}\left(\xi \right)=(1-|\xi {|}^2/R^2{)}_{+}^{\lambda }\widehat{f}\left(\xi \right)$. E. Stein conjectured$$\parallel {\left(\sum _j|T_{R_j}^{\lambda }{f}_i{|}^2\right)}^{1/2}{\parallel }_p\le C\parallel {\left(\sum _j|f_j{|}^2\right)}^{1/2}{\parallel }_p$$for $\lambda \&gt;0$, $\frac{4}{3}\le p\le 4$ and $C=C_{\lambda ,p}$. We prove this conjecture. We prove also $f\left(x\right)={lim}_{j\to \infty }{T}_{2^j}^{\lambda }f\left(x\right)$ a.e. We only assume $\frac{4}{3+2\lambda }\&lt;p\&lt;\frac{4}{1-2\lambda }$.