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In this note we study the relation between oscillatory integral operators and convolution operators, and also the sharpness of L p L^p -estimates for singular oscillatory integral operators.

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

We prove a Covering Theorem that allows us to prove modified norm inequalities for general maximal operators. We will also give applications to convergence of a sequence of linear operators and the differentiation of the integral.

If E E and F F are Banach lattices such that E E is separable and F F has the countable interpolation property, then the space of all continuous regular operators L \mathcal {L} r ( E , F ) ^r(E,F) has the Riesz decomposition property. This result is a positive answer to a conjecture posed by A. W. Wickstead.

We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on Lorentz spaces follows from the appropriate norm estimates.

We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on L p L^{p} of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces.

This paper discusses the properties of a class of p p -adic Walsh convolution operators. The class consists of those 1 1 -parameter sets of operators with kernels that can be represented as the p p -adic Walsh-Fourier integral of a uniformly quasi-convex function. The paper proves that the maximal operators associated with each 1 1 -parameter set are all of strong type ( ∞ , ∞ ) (\infty ,\infty ) and of weak type ( 1 , 1 ) (1,1) .

A condition for the boundedness of a Calderón-Zygmund operator on a new class of Banach function spaces is studied.

In this paper we are concerned with linear operators K : L → M K:L \to M , where L is a Riesz subspace of measurable, finite a.e. functions and M is the class of all measurable, finite a.e. functions defined by k ( x , y ) k(x,y) is a measurable kernel. It will be shown that the class I [ L , M ] I[L,M] of all such integral operators is a Dedekind complete Riesz space, an ideal and a band in the space of order bounded linear maps T : L → M T:L \to M .

In this paper continuous extensions of the Lebesgue Integral which integrate their almost everywhere approximate derivatives are considered. First, those extensions which are generated by a single function are characterized. Then the largest extension which is contained in all maximal extensions is considered and shown to contain the wide sense Denjoy Integral. Finally, properties are given which characterize this largest extension.

We consider operators of the form∫−∞∞Ftϕ(t)dt\int _{-\infty }^\infty {{F_t}} \phi \left ( t \right )dtwhereFt{F_t}is a11-parameter family of Fourier integral operators andϕ(t)dt\phi \left ( t \right )dta tempered distribution on the real line. We extend the result given in [1].

An operator means a bounded linear operator on a complex Hilbert space. The familiar Fuglede-Putnam theorem asserts that if A and B are normal operators and if X is an operator such that A X = X B AX = XB , then A ∗ X = X B ∗ {A^ \ast }X = X{B^ \ast } . We shall relax the normality in the hypotheses on A and B. Theorem 1. If A and B ∗ {B^\ast } are subnormal and if X is an operator such that A X = X B AX = XB , then A ∗ X = X B ∗ {A^ \ast }X = X{B^ \ast } . Theorem 2. Suppose A, B, X are operators in the Hilbert space H such that A X = X B AX = XB . Assume also that X is an operator of Hilbert-Schmidt class. Then A ∗ X = X B ∗ {A^ \ast }X = X{B^ \ast } under any one of the following hypotheses: (i) A is k-quasihyponormal and B ∗ {B^ \ast } is invertible hyponormal, (ii) A is quasihyponormal and B ∗ {B^\ast } is invertible hyponormal, (iii) A is nilpotent and B ∗ {B^\ast } is invertible hyponormal.

The purpose of this paper is to extend an interpolation theorem for sublinear operators on Lp spaces, obtained by Calderon and Zygmund, to similar operators acting on Hp spaces, p >0 [I]. We begin by making a few definitions. Let T be a mapping defined on a class of functions that are analytic in the interior of the unit circle I w| 0, if the domain of T includes Ha and there exists a constant M, independent of F in Ha, such that jjTFjjb

with condition (4) satisfied, and similar convolution transforms. The object of this paper is to generalize the form of G(t) by replacing the exponential factor in the integral on the right of (2) by generalized Fourier kernels [4, chap. 8]. Owing to the complexity of the Presented to the Society, September 3, 1953; received by the editors February 18, 1953 and, in revised form, April 8, 1953. 880

Let ( w 1 , w 2 , w 3 ) ({w_1},{w_2},{w_3}) be an ordered triple of real numbers such that w 1 + w 2 + w 3 = 1 {w_1} + {w_2} + {w_3} = 1 . Let g be a real-valued function on the entire real axis which is of bounded variation on every closed interval. For f a real-valued function on the entire real axis which is bounded on a closed interval [a, b], we use the F. Riesz step function approach to define the concept of f being ( w 1 , w 2 , w 3 ) ({w_1},{w_2},{w_3}) g-summable over [a, b], and we define the integral \[ [ F , ( w 1 , w 2 , w 3 ) ] s ∫ a b f ( x ) d g ( x ) [F,({w_1},{w_2},{w_3})]s\int _a^b {f(x)dg(x)} \] when f has this property. We show that this integral extends the weighted refinement integral [ F , ( w 1 , w 2 , w 3 ) ] ∫ a b f ( x ) d g ( x ) [F,({w_1},{w_2},{w_3})]\int _a^b {f(x)dg(x)} for f’s as above. This paper generalizes the method of Pasquale Porcelli for the Stieltjes mean sigma integral. We present an existence theorem for the integral defined here involving saltus and continuous parts of g. We establish a convergence theorem for this integral which is analogous to the Lebesgue Dominated Convergence Theorem for the Lebesgue-Stieltjes integral.

The purpose of this note is to settle a problem posed by Caradus, Pfaffenberger, and Yood; namely, it is proved that every Riesz operator R on a Hilbert space has a decomposition R = C + Q where C is compact and both Q and CQ QC are quasinilpotent. Let H denote a complex, separable, infinite dimensional Hilbert space. In [5], West showed that every Riesz operator was a decomposable Riesz operator, i.e., R = C + Q where C is a compact operator and Q is quasinilpotent. In general, this decomposition is not unique. A Riesz operator is said to be fully decomposable if R is decomposable and, in addition, C commutes with Q for some decomposition C and Q. In [1, p. 58], an example of Gillespie and West was given showing that there are some Riesz operators on H which are not fully decomposable. They produced a Riesz operator R for which no decomposition could commute. This leads to the following question proposed in [1, p. 59]: Can every Riesz operator be decomposed in such a manner that the commutator CQ QC is quasinilpotent? The purpose of this note is to give a positive answer to this question, and, in fact, a slightly stronger result is proved. The key to our proof is a lemma of Gohberg and Krein which was stated without proof in [3, p. 17] and was later stated and proved by Stampfli [4]. LEMMA 1 [GOHBERG-KREIN, STAMPFLI]. Let E be a closed set in C. Let a(T) \ E consist of isolated points (A>} which of necessity cluster only on E. Let each X be a point of finite multiplicity. Then, T = S + K where K is compact and a(S) c E. Our theorem will depend heavily on the "Stampfli decomposition" and on its notation. Let us recall the pertinent steps. It was shown by Stampfli [4] that for a T satisfying the hypotheses of Lemma 1, Received by the editors March 5, 1976. AMS (MOS) subject classifications (1970). Primary 47B05.

We show that, on local fields, "nice" singular integral transforms preserve HP-spaces for 0 1 (i.e., 7r(pkx) = 7T(x) for all k 6 Z and ir is constant on ?T but not on h-l). v is, thus, constant on cosets of T-m+h in ?-m\m+l =lx e K: Ixl = qml and f1m (x)dx 0 for all m e Z. Denote Q(x) c7T1b(x)/lxl) where c 7T = 1/1(r). (See [5] for details.) Distributions on K have been identified with regular functions on K x Z. The regularization kernel R k(x) = qk _k (x), where (F k is the Received by the editors February 27, 1974. AMS (MOS) subject classifications (1970). Primary 44A25, 12B99, 46J 15; Secondary 43A70.