It is known that Gabor expansions do not converge unconditionally in and that cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in -norm.
We obtain interpolation inequalities for derivatives: , and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ)where is the Orlicz norm relative to the function . The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher...
We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.
This note contains a simple example which does clearly indicate the differences in the Henstock-Kurzweil, McShane and strong McShane integrals for Banach space valued functions.