Bounded linear operators Interpolation of Banach spaces Integral operators on $L^p$ spaces Bergman spaces Bloch and Besov spaces The Berezin transform Toeplitz operators on the Bergman space Hankel operators on the Bergman space Hardy spaces and BMO Hankel operators on the Hardy space Composition operators Bibliography Index.

http://www.gbv.de/dms/goettingen/52319126X.pdf

Operator theory in function spaces

A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1 <p, q, n/X < x. When p =q' or p = 2 or q = 2, f and N are explicitly evaluated. A maximizing f is also shown to exist for other inequalities: (i) The Okikiolu, Glaser, Martin, Grosse, Thirring inequality: Kn vjj Vf 11?2 11 IIx-bfjIp, n ? 3, 0 < b < 1, p = 2n/(2b + n - 2). (This was known before, but the proof here has certain simplifications.) (ii) The doubly weighted HLS inequality of Stein and Weiss:

Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

It is clear that for given I,un } and t, the better theorem of this kind would be the one in which (2) is proved for the larger class of functions f. In this paper we shall show that certain known "invariance principles" can under some hypotheses be improved by considerably enlarging the class of functions for which (2) holds. This will be done by considering spaces S other than the customary ones. For example, in studying convergence to the Wiener process, it is usual to let S be the space (denoted e) of continuous functions with the uniform topology. However, this choice does not fully exploit the pleasant properties of the Wiener path-functions, which are not only continuous but also Holder continuous of any order up to 1/2. Therefore we shall attempt to use spaces Lip5 in place of e as the function-space S. When weak convergence can be established using such spaces, the class of functionals for which (2) is known to hold becomes much larger than before. To carry out the idea sketched above it is necessary to have a criterion which guarantees that the sample functions of a stochastic process are a.s.

/pdf/on-convergence-of-stochastic-processes-25ffw9tvsu.pdf

On convergence of stochastic processes

n(r, F) as the number of poles in ]z] d r and = f r n(t, F) dt N(r, F) J0 $ Then T(r, F) = re(r, F) + N(r, F) is called the Nevanlinna characteristic function of F(z). The function T(r, F ) i s convex increasing function of log r, so tha t T(1, F) = lim T(r, F) r--~l always exists as a finite or infinite limit. I f T(1, F) is finite we say tha t F(z) has bounded characteristic in ]z] < 1. Examples show tha t F(z) m a y have bounded characteristic in ]z] < 1, even i f / (z) does not.(1) We may take for instance /(z) to be a regular function

/pdf/on-the-characteristic-of-functions-meromorphic-in-the-unit-53gisr96lg.pdf

On the characteristic of functions meromorphic in the unit disk and of their integrals

The principal problem considered is the determination of all nonnegative functions, V(x), such that IIhTf(x)V(x)14 < ClIf(x)V(x)lIp where the functions are defined on R', 0 < y < n, 1 < p < n/y, l/q = l/p -yn, C is a constant independent of f and Tyf (x) = S f(x y) Ly|f dy. The main result is that V(x) is such a function if and only if (-fQ [V(X)]qdx (jff [vx) P'd)1 (IQIJQVX]^ (QI J [V(X)]-p, ) where Q is any n dimensional cube, IQI denotes the measure of Q, p' = p/(p 1) and K is a constant independent of Q. Substitute results for the cases p = 1 and q = oo and a weighted version of the Sobolev imbedding theorem are also proved.

/pdf/weighted-norm-inequalities-for-fractional-integrals-3hiom7ujpd.pdf

J. E. Littlewood

Papers

Some properties of fractional integrals II