Weighted norm inequalities for convolution and Riesz potential

In this paper, we prove analogues of O'Neil's inequalities for the convolution in the weighted Lebesgue spaces. We also establish the weighted two-sided norm inequalities for the potential operator

Fourier inequalities in Morrey and Campanato spaces

We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} \|\widehat f\|_{X_{p,q}^\lambda} \lesssim \|f\|_{Y}, \end{equation} where $X$ is either a Morrey or Campanato space and $Y$ is an appropriate function space. In the case of the Morrey space we sharpen the estimate $\|\widehat f\|_{M_{p,q}^\lambda} \lesssim \|f\|_{L_{s',q}},$ $s\geq 2,$ $\frac{1}{s} = \frac{1}{p}-\frac{\lambda}{n}.$ We also show that \eqref{introduction} does not hold when both $X$ and $Y$ are Morrey spaces. If $X$ is a Campanato space, we prove that \eqref{introduction} holds for $Y$ being the truncated Lebesgue space

Interpolation methods for stochastic processes spaces

In this paper the scales of classes of stochastic processes are introduced. New interpolation theorems and boundedness of some transforms of stochastic processes are proved. Interpolation method for generously-monotonous processes is entered. Conditions and statements of interpolation theorems concern the xed stochastic process, which di ers from the classical results

Net spaces and boundedness of integral operators

In this paper we introduce new functional spaces which we call the net spaces. Using their properties, the necessary and sufficient conditions for the integral operators to be of strong or weak-type are obtained. The estimates of the norm of the convolution operator in weighted Lebesgue spaces are presented

Convolution inequalities in Lorentz spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. In particular, we obtain the limit case of the Young-O'Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces. Finally, norm inequalities for the potential operator are presented

A sharp Remez inequality for trigonometric polynomials

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Norm convolution inequalities in Lebesgue spaces

We obtain upper and lower estimates of the (p; q) norm of the con-volution operator. The upper estimate sharpens the Young-type inequalities due to O'Neil and Stepanov