We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

In this paper we consider products of composition and differentiation operators on the Hardy spaces. We provide a complete characterization of boundedness and compactness of these operators. M. Moradi and M. Fatehi [1] obtain the explicit condition for these operators to be Hilbert-Schmidt operators. We have a theoretical application on the composition operators with series of perfect symbols.

As for the affine energy, Edir Junior and Ferreira Leite establish the existence of minimizers for particular restricted subcritical and critical variational issues on BV(Ω). Similar functionals exhibit deeper weak* topological traits including lower semicontinuity and affine compactness, and their geometry is non-coercive. Our work also proves the result that extremal functions exist for certain affine Poincaré-Sobolev inequalities.

The Orlicz ( \(\ell\)2,\(\ell\)1)-mixed inequality of integers and fractional dimensions who states that, with a bit of extend, for all sequences of bilinear forms AL: \(\mathbb{K}\)n x \(\mathbb{K}\)n  \(\rightarrow\) \(\mathbb{K}\) and all positive integers n, where  \(\mathbb{K}\)n denotes  \(\mathbb{R}\)n or  \(\mathbb{C}\)n endowed with the supremum norm. For that we follow D.Núñez-Alarcón, D. Pellegrino, and D. Serrano-Rodríguez [1]] to extend this inequality to series of multilinear forms, with \(\mathbb{K}\)n endowed with \(\ell\)1+ \(\epsilon\) norms for all successive gradually of the general 0 ≤ ϵ ≤ ∞.