On the general net spaces

Abstract

The theory of the net spaces has a lot of applications in harmonic analysis and approximation theory. By the interpolation theorems of the net spaces it was obtained new results in many classical problems of analysis. In particular, new estimates of norms of the convolution operator were proved in the terms of the net spaces. Also, in the context of the theory of net spaces were considered questions related to the Fourier multipliers, estimates of norms of the Fourier transforms (Hardy and Littlewoods’s theorems), inequalities in various metrics and other problems of analysis. In this paper are introduced general net spaces Np;q; depending on three positive parameters. Introduced spaces is close to the known net spaces. In paper we study properties of Np;q; spaces. In particular, we study their interpolation properties. Besides that we give equivalent norms in these spaces. Also we show that these spaces increase by the second parameter q. In paper we give some applications of general net spaces. We obtain estimate of norm of the Fourier transform of function from the Lorentz space. Analogous estimates were proved earlier for the net spaces and were used for proving criteria of integrability of Fourier transforms.