John J. F. Fournier

Mixed norms and rearrangements: Sobolev's inequality and Littlewood's inequality

SummaryIn the 1930s, J. E. Littlewood and S. L. Sobolev each found useful estimates for Lp-norms. These results are usually not regarded as similar, because one of them is set in a discrete context and the other in a continuous setting. We show, however, that certain basic facts about mixed norms can be used to simplify proofs of both of these estimates. The same method yields a proof of a form of the isoperimetric inequality. We consider the effect of measure-preserving rearrangement on certain sums of permuted mixed norms of functions on RK, and show these sums are minimal when the rearranged function, f∼ say, has the property that, for each positive real number λ, the set on which ¦f∼¦>λ is a cube with edges parallel to the coordinate axes. Finally, we use the fact about rearrangements to prove sharper forms of the estimates of Littlewood and Sobolev.

Cone conditions and properties of Sobolev spaces

Abstract The Sobolev imbedding theorem and certain interpolation inequalities for Sobolev spaces are established for a wider class of domains than has been covered by earlier proofs. This class is defined by a weakened, measure theoretic version of the cone condition. The proofs are elementary.

Inclusions and noninclusion of spaces of convolution operators

Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type (p, q) by Lp(G). P It is shown that, if lllq 1/21 2, but it is shown that if p # 1, 2, c, then there exists a convolution operator that is of restricted weak type (p, p) but is not of weak type (p, p). Many of these results also hold for the spaces of operators that commute with left translation rather than right translation. Further refinements are presented in three appendices.

On the Hausdorff-Young theorem for amalgams

Certain function spaces called amalgams have been used and studied in several recent papers on abstract harmonic analysis. In this paper, we give a new proof of a Hausdorff-Young theorem for amalgams on locally compact abelian groups. We also prove some complementary results about amalgams and their Fourier transforms, and in particular give simple proofs of some facts about the Fourier multipliers from certain spaces of functions with compact support intoA(G).

Some sufficient conditions for uniform convergence of Fourier series

Abstract We show that if a function ƒ on the unit circle belongs to the Besov space Λ( 1 p , p, 1) , where p is an index in the interval (2, ∞), then ƒ belongs to the Figa-Talamanca space A p . It follows that if ƒ ϵ Λ( 1 p , p, 1) then the Fourier series of ƒ converges uniformly. The latter implication was proved earlier by Garsia, using a rearrangement inequality proved by combinatorial methods. Our proof uses only well-known notions in harmonic analysis. We also study the connection between these results on uniform convergence and some conditions for the integrability of sums of trigonometric series.

Majorants andL p norms

It is shown that ifG is an infinite compact abelian group, thenLp (G) has the upper majorant property only ifp is even orp=∞.

An Integrability Theorem for Unbounded Vilenkin Systems

We consider Vilenkin systems (n) 1 and series P 1=0 ann with coecients tending to 0. We suppose that the coecients satisfy the regularity condition that

Smoothness conditions and integrability theorems on bounded Vilenkin groups

Various criteria, in terms of forward differences and related operations on coefficients, are shown to imply that certain series on bounded Vilenkin groups represent integrable functions. These results include analogues of known integrability theorems for trigonometric series. The method of proof is to pass from the given series to a derived series, and to deduce the integrability of the original series from smoothness properties of the latter.

Random Fourier–Stieltjes transforms on partially ordered ǵroups

If one changes at random the signs of the Fourier coefficients of an L2 function the result is still the sequence of Fourier coefficients of an L2 function. L2 is the only LP space with this property; in fact if a sequence remains a Fourier transform for every insertion of ± signs then it is really the transform of an L2 function ((6), p. 215, Theorem 8·14). Our main theorem is a version of this principle, for sequences defined on suitable subsets of a group; it is phrased in terms of large sets of choices of ± signs rather than all sequences of ± signs.