using a generalized spherical mean operator, we obtain a generalization of titchmarsh&apos;s theorem for the dunkl transform for functions satisfying the (&apos;; p)-dunkl lipschitz condition in the space lp(rd;wl(x)dx), 1 < p 6 2, where wl is a weight function invariant under the action of an associated re ection group.

For a space (X, d, μ) of homogeneous type and fractional integral operator Kα defined on we find necessary sufficient condition the exponent q governing compactness from Lp(X) to Lq(X), where 1 ≤ p, < ∞ μ(X) ∞.

If 0 < p < 1 we classify completely the linear operators T: Lp -X where X is a p-convex symmetric quasi-Banach function space. We also show that if T: LLo is a nonzero linear operator, then forp < q < 2 there is a subspace Z of Lp, isomorphic to Lq, such that the restriction of T to Z is an isomorphism. On the other hand, we show that if p < q < o, the Lorentz space L(p, q) is a quotient of Lp ...

We show that a suitable weak solution to the incompressible Navier–Stokes equations on R3×(?1,1) is regular R3×(?1,0] if ?3u belongs M2p/(2p?3),?((?1,0);Lp(R3)) for any ?>1 and p?(3/2,?), which logarithmic-type variation of Morrey space in time. For each this is, up logarithm, critical with respect scaling equations, contains all spaces Lq((?1,0);Lp(R3)) are subcritical, 2/q+3/p<2.

We give a direct proof of the ‘upper’ Khintchine inequality for a noncommutative symmetric (quasi-)Banach function space with nontrivial upper Boyd index. This settles an open question of C. Le Merdy and the fourth named author [24]. We apply this result to derive a version of Rosenthal’s theorem for sums of independent random variables in a noncommutative symmetric space. As a result we obtain...

This paper is concerned with Schrödinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the Lp-Lq estimate of solution operator in free case. This estimate, combining with the results of fractionally integrated groups, allows us to further obtain the Lp estimate of solutions for the initial data belonging to a dense sub...

Using elementary arguments based on the Fourier transform we prove that for 1 ≤ q < p < ∞ and s ≥ 0 with s > n(1/2 − 1/p), if f ∈ L(R) ∩ Ḣ(R) then f ∈ L(R) and there exists a constant cp,q,s such that ‖f‖Lp ≤ cp,q,s‖f‖ θ Lq,∞‖f‖ 1−θ Ḣs , where 1/p = θ/q+(1− θ)(1/2− s/n). In particular, in R we obtain the generalised Ladyzhenskaya inequality ‖f‖L4 ≤ c‖f‖ 1/2 L2,∞ ‖f‖ 1/2 Ḣ1 . We also show that f...