The bilinear maximal operator defined below maps Lp × Lq into Lr provided 1 < p, q < ∞, 1/p + 1/q = 1/r and 2/3 < r ≤ 1. Mfg(x) = sup t>0 1 2t ∫ t −t |f(x+ y)g(x− y)| dy. In particular Mfg is integrable if f and g are square integrable, answering a conjecture posed by Alberto Calderón. 1. Principal results In 1964 Alberto Calderón defined a family of maximal operators by Mfg(x) = sup t>0 1 2t ∫...

so ‖f‖p ≤ ‖f‖q · μ(X) 1 p − 1 q . Hence if f is in Lq, the left-hand side is finite hence so is the right-hand side, so f is in Lp. Also, the inequality shows that if ‖f‖p is small then ‖f‖q is also small, hence the inclusion Lq ↪→ Lp is continuous 2. Let X ⊂ Pn be an irreducible projective variety of dimension k, G(`, n) the Grassmannian of `-planes in Pn for some ` < n− k, and C(X) ⊂ G(`, n) ...

A new majorization-minimization framework for lp-lq image restoration is presented. The solution is sought in a generalized Krylov subspace that is build up during the solution process. Proof of convergence to a stationary point of the minimized lp-lq functional is provided for both convex and nonconvex problems. Computed examples illustrate that high-quality restorations can be determined with...

In this paper we establish a discrete Calderón’s identity which converges in both Lq(Rn+m) (1 < q <∞) and Hardy space H(R ×Rm) (0 <p ≤ 1). Based on this identity, we derive a new atomic decomposition into (p, q)-atoms (1 < q <∞) on H(R × R) for 0 < p ≤ 1. As an application, we prove that an operator T , which is bounded on Lq(Rn+m) for some 1 < q <∞, is bounded from Hp(Rn×Rm) to Lp(Rn+m) if and...

We examine the stochastic parabolic integral equation of convolution type U(t) +A ∫ t 0 k1(t− s)U(s) ds = ∫ t 0 k2(t− s)G(s) dWH(s), t ≥ 0, where U(t) takes values in Lq(O;R) with O a σ-finite measure space, and q ∈ [2,∞). The linear operator A maps D(A) ⊂ Lq(O;R) into Lq(O;R), is nonnegative and admits a bounded H∞-calculus on Lq(O;R). The kernels are powers of t, with k1(t) = 1 Γ(α) t α−1, k2...

A,(T) = [L9(T) ® LQ(T)]IK where K is the kernel of the convolution operator c:Lp(S>LQ(T)->C0(T) by (f®g)(y)(f* g)(y)-4*(r) is the /?-Fourier algebra which was introduced by Figa-Talamanca in [6] where it was shown that ^ ( T ) * isisometrically isomorphic to MP(T), the bounded, translation invariant, linear operators on LP(T), Herz [11] showed that AP(V) is a Banach algebra under pointwise mult...

This paper presents a new efficient approach for the solution of the lp-lq minimization problem based on the application of successive orthogonal projections onto generalized Krylov subspaces of increasing dimension. The subspaces are generated according to the iteratively reweighted least-squares strategy for the approximation of lp/lq-norms by weighted l2-norms. Computed image restoration exa...

A classical theorem of Hausdorff-Young shows that when 1 < p < 2, the system of equations rp(n) = cn (-oo < n < oo) admits a solution <p in Lq{~tt,<tt) whenever {c„} E /', Here, as usual, <p denotes the complex Fourier transform of <p and q is the conjugate exponent given by p~l + q~l = 1. The purpose of this note is to show that if a set {Xn} of real or complex numbers is "sufficiently close" ...