Multilinear interpolation between adjoint operators

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].

One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form rn for some n>0. The aim of this paper is to study the boundedness of Calderon–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the Lp spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (ℝn,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytonen (2011) and we are able to obtain quite a few properties similar to those of Calderon–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L1, boundedness from L∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderon–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderon–Zygmund operators and BMO functions.

This paper deals with multilinear operators acting in products of Banach spaces that factor through a canonical mapping. We prove some factorization theorems and characterizations by means of norm inequalities for multilinear operators defined on the n-fold Cartesian product of the space of bounded Borel measurable functions, respectively, products of Banach function spaces. These factorizations allow us to obtain integral dominations and lattice geometric properties and to present integral representations for abstract classes of multilinear operators. Finally, bringing together these ideas, some applications are shown, regarding for example summability properties and representation of multilinear maps as orthogonally additive n-homogeneous polynomials.

In the present chapter we examine the connections between functional calculus and interpolation spaces. As an ‘appetiser’, in Section 6.1 we present two central ideas: a model describing the real interpolation spaces (X, D(A)) θ,p using the functional calculus and a theorem of Dore. In Section 6.2 we examine the first of these ideas, proving several representation results for the spaces (X, D(Aα)) θ,p . Then we introduce extrapolation spaces for injective operators (Section 6.3). With the help of these spaces in Section 6.4 we derive two fundamental results (Theorem 6.4.2 and Theorem 6.4.5) that lead to more characterisations of interpolation spaces by functional calculus (Section 6.5.1) and a generalisation of Dore’s theorem (Section 6.5.3). In Section 6.6 we establish all the common properties of fractional domain spaces as intermediate spaces: density, the moment inequality, reiteration. Moreover, we prove the intriguing fact that for an operator A ∈ BIP(X) the fractional domain spaces equal the complex interpolation spaces (Theorem 6.6.9). Finally we characterise growth conditions like supt≥1 ‖tθ B(t + A)−1‖ < ∞ in terms of interpolation spaces (Section 6.7).

The problem of creation and investigation of parallel Monte Carlo algorithms for solving LargeScale Air Pollution Problems is studied. We consider a model of Air pollution transport. This model includes diffusion, advection (drift due to the wind), deposition and chemical reactions. For obtaining the initial and boundary conditions of the model it is necessary to deal with Noisy Data. The work consists of three parts. In the first part we obtain a practical efficient Monte Carlo method for estimating the optimum amount of smoothing from the noisy data. In the second part a special Monte Carlo techniques (Monte Carlo with frozen velocities) for calculation of concentrations is developed and studied. An error analysis of this method is realized. The last part of the work deals with parallel realization of Monte Carlo methods on distributed transputer system. A numerical results for estimation of the Speed-Up and Efficiency are obtained. INTRODUCTION Monte Carlo methods are a powerful tool in many fields of mathematics and engineering. It is known that these methods have proved to be very efficient in solving multidimensional problems in composite domains [1,2,3]. Moreover, it is shown that for some problems(including one dimensional problems) in the corresponding functional spaces Monte Carlo methods have better speed of convergence than the optimal deterministic methods in such functional spaces [4,5]. * Supported by the Ministry of Science and Education of Bulgaria, Grand I 210 /1992. Transactions on Information and Communications Technologies vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 480 Applications of Supercomputers in Engineering An important advantage of these algorithms is that Monte Carlo ones allow to find directly the unknown functional of the solution of the problem with a number of operations, necessary to solve the problem in one point of the domain [1,6]. 1 PRELIMINARIES Our approach is based on an integral representation of the solution of the problem. The Fredholm integral equation t u(r, t) = jdtj k(r', f; r, t)u(r', t')dr' + f(r, t), (1) 0 G where re G C R*; t e [0,T]or (x,x')u(x')dx' + f(r,t), (2) D (« = Ku + /) where x = (r, t) £ D = G x [Q,T] and K is an integral operator, is studied. It is assumed that u, / G L, K G [L — •> L], where Lisa Banach space of integrable functions. Monte Carlo method for evaluation of linear functionals of the solution of the following type /t(z)w(z)dz = (12, A) (3) D is considered. The adjoint equation %* = JT w* 4k (4) will be used. In (4) w*, h G L*, K* G [L* — » L*], L* is the adjoint functional space and K* is an adjoint operator. For many applications L = LI and II / lk= / I /(*) I dx; D \\K\\L,<su.p f\k(x,x')\dx' (5) X J D In the case h(x) G Z/oo, sence L\* = L^ and Transactions on Information and Communications Technologies vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3517 Applications of Supercomputers in Engineering 481 || A||z_=sup|/t(z) |,zED. (6) It is assumed that || K* ||< 1,where n is a natural number, because the studied method uses the Neuman series CO « = £#'/• (7) i=0 But the condition || KTM ||< 1 is not very strong, since as shown by K.Sabelfeld [7] it is possible to construct Monte Carlo method when von Neuman series does not converge. Using method of analytical extension of resolvent through change of spectral parameter [7,p.60] it is possible to create convergent method when von Neuman series does not converge or to accelerate the rate of convergence when it converges slowly. It is easy to show that J = (u,h) = (/,«•). (8) So, consider Monte Carlo method for evaluation of functional (8).(One can see that when fc(x,xi) = 0we have a problem of evaluation of integrals.) Let £ £ D be a random point with a density p(x) and let there be n realizations of the random point £,-(i — 1,2, ...,ra). Let a random variable #(£) be defined in D, such that E6(t) = J Then the computational problem becomes one of calculating repeated realizations of 9 and of combining them into an appropriate statistical estimator of J. Because of the nature of the process every realization of 0 is a Markov chain. So, next we will use the Markov chain as a synonym of the realization of a random variable of this type. As approximate value of the linear functional J is set up

In this paper, we apply function parameters, introduced by Persson, to real interpolation of Lorentz martingale spaces. Some new interpolation theorems concerning Lorentz martingale spaces are formulated. The results that we obtain generalize some fundamental interpolation theorems in classical martingale Hp theory.

Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions

In this paper, we study the K-envelopes of the real interpolation methods with function space parameters in the sense of Brudnyi and Kruglyak [Y. A. Brudnyi and N. Ja. Kruglyak, Interpolation functors and interpolation spaces (North-Holland, Amsterdam, Netherlands, 1991)]. We estimate the upper bounds of the K-envelopes and the interpolation norms of bounded operators for the KΦ-methods in terms of the fundamental function of the rearrangement invariant space related to the function space parameter Φ. The results concerning the quasi-power parameters and the growth/continuity envelopes in function spaces are obtained.

Geophysical Inverse Theory and Regularization Problems

Interpolation in spaces of measurable functions

We study some new summability properties of multilinear operators. We introduce the concepts of φ‐summing, φ semi‐integral and φ‐dominated multilinear maps generated by Orlicz functions. We prove a variant of Pietsch's domination theorem for φ‐summing operators, providing also a characterization of φ‐dominated operators in terms of factorizations. We analyze vector‐valued inequalities associated to these maps, which are applied to obtain general variants of multiple summing operators. We also study translation invariant multilinear operators acting in products of spaces of continuous functions, proving that a factorization theorem can be obtained for them as a consequence of a suitable representation of the corresponding normalized Haar measure.

Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exists which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In this Lecture Note, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.

In this paper we characterize the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ of the variable exponent Hormander space $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ when the exponent $$p(\cdot )$$ satisfies the conditions $$0 < p^- \le p^+ \le 1$$ , the Hardy-Littlewood maximal operator $$M$$ is bounded on $$L_{p(\cdot )/p_0}$$ for some $$0 < p_0 < p^-$$ and $$\Omega $$ is an open set in $$\mathbb {R}^n$$ . It is shown that the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ is isomorphic to the Hormander space $$\mathcal {B}^{\mathrm {loc}}_\infty (\Omega )$$ (this is the $$p^+ \le 1$$ counterpart of the isomorphism $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq \mathcal {B}^{\mathrm {loc}}_{\widetilde{p'(\cdot )}} (\Omega )$$ , $$1 < p^- \le p^+ < \infty $$ , recently proved by the authors) and hence the representation theorem $$\bigl ( \mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq l^{\mathbb {N}}_\infty $$ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $$p(\cdot ) \equiv p$$ , $$0 < p < 1$$ , are also given (e.g. $$\mathcal {B}^c_p (\Omega )$$ does not contain any infinite-dimensional $$q$$ -Banach subspace with $$p < q \le 1$$ or the quasi-Banach space $$\mathcal {B}_p \cap \mathcal {E}'(Q)$$ contains a copy of $$l_p$$ when $$Q$$ is a cube in $$\mathbb {R}^n$$ ). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent Hormander spaces is proposed.

Optimal Birkhoff Interpolation and Birkhoff Numbers in Some Function Spaces

This chapter concludes the analysis of the tracial setting by showing that real interpolation extends to the context of tracial Lp and Orlicz spaces in a manner which closely mimics the classical theory. After a brief consideration of a noncommutative Marcinkiewicz interpolation theorem, we then go on to enlarge our pool of noncommutative spaces by using the K-method of interpolation to introduce a very general class of noncommutative rearrangement invariant Banach function spaces – the so-called monotone Riesz-Fischer spaces. Our approach to this by now well-known theory is slightly non-standard in that we introduce the theory by exploiting the known theory of modular interpolation spaces. As a result we are able to get to the cut of the chase a lot quicker.