Généralisation des algèbres de Beurling.

*(English)*Zbl 0531.46039
Publ. Math. Orsay 83-05, 113 p. (1983).

The author introduces the notion of generalized Beurling algebras and studies, among other things, multipliers and pseudodifferential operators related to them. Let w(x) be a positive function on \(R^ N\) satisfying the following conditions: (a) \(w(x+y)\leq Cw(x)w(y),\) (b) \(w^{-1}\) is integrable, (c) \((w^{-1})*(w^{-1})\leq Cw^{-1}\), and (d) \(1\leq w(x)\leq C(1+| x|^ 2)^ n,\) where C is a positive constant and n is a positive integer. We denote by A the space of all functions f in \(L^ 2(R^ N)\) with \(\int | \hat f|^ 2wdx<\infty\), where \(\hat f\) is the Fourier transform of f. Then, \(A_ w\) is seen to be an algebra under the pointwise operations and is called a Beurling algebra after A. Beurling’s pioneering work [Ann. Inst. Fourier 14(1964), 1-32 (1965; Zbl 0133.075)]. Recently, R. R. Coifman and Y. Meyer [Astérisque 57, 185 p. (1978; Zbl 0483.35082) Ch. I] used such algebras to describe \(L^ 2\)-boundedness of certain pseudodifferential operators on \(R^ N.\)

In the paper under review, the author wants to see how much one can weaken the hypothesis (d) without losing main properties of Beurling algebras including Coifman and Meyer’s. In fact, he considers a general weight w(x), which is just a locally integrable positive function on \(R^ N\), and define \(A_ w\) roughly as follows: \(A_ w=\{f\in {\mathcal S}'(R^ N):\int | \hat f(\xi)|^ 2w(\xi)d\xi<\infty \}\) with norm \(\| f\| =(\int | \hat f(\xi)|^ 2w(\xi)d\xi)^{1/2}\). Such an \(A_ w\) is called a generalized Beurling algebra if it is a Banach algebra and an \(L^ 2\)-puzzle at the same time. Here, an \(L^ 2\)-puzzle E is, again speaking very roughly, a translation invariant Banach space whose elements are generalized functions on \(R^ N\) and which admits sufficiently many multipliers \(\Delta\) (x) of special kind so that, for each \(f\in E\), the norm \(\| f\|\) is equivalent to \((\int \| f(x)\Delta(x-t)\|^ 2dt)^{1/2}.\)

The paper begins with basic properties of \(L^ p\)-puzzles. In the second section he obtains conditions on weights w(x) which imply that \(A_ w\) are \(L^ 2\)-puzzles. The author obtains a result on continuity of pseudodifferential operators defined on \(A_ w\), which is an \(L^ 2\)- puzzle. The third section deals with conditions for \(A_ w\) to be an algebra and with those which make an \(A_ w\) an \(L^ 2\)-puzzle when it is an algebra. In the fourth section a sufficient condition is given for a pseudodifferential operator on a generalized Beurling algebra to be continuous. Multipliers are studied in detail for two special classes of examples. In the final section he discusses certain weighted \(L^ 2\)- spaces which appear to be the Fourier transform of the spaces of the form \(A_ w\). An application to BMO is described.

In the paper under review, the author wants to see how much one can weaken the hypothesis (d) without losing main properties of Beurling algebras including Coifman and Meyer’s. In fact, he considers a general weight w(x), which is just a locally integrable positive function on \(R^ N\), and define \(A_ w\) roughly as follows: \(A_ w=\{f\in {\mathcal S}'(R^ N):\int | \hat f(\xi)|^ 2w(\xi)d\xi<\infty \}\) with norm \(\| f\| =(\int | \hat f(\xi)|^ 2w(\xi)d\xi)^{1/2}\). Such an \(A_ w\) is called a generalized Beurling algebra if it is a Banach algebra and an \(L^ 2\)-puzzle at the same time. Here, an \(L^ 2\)-puzzle E is, again speaking very roughly, a translation invariant Banach space whose elements are generalized functions on \(R^ N\) and which admits sufficiently many multipliers \(\Delta\) (x) of special kind so that, for each \(f\in E\), the norm \(\| f\|\) is equivalent to \((\int \| f(x)\Delta(x-t)\|^ 2dt)^{1/2}.\)

The paper begins with basic properties of \(L^ p\)-puzzles. In the second section he obtains conditions on weights w(x) which imply that \(A_ w\) are \(L^ 2\)-puzzles. The author obtains a result on continuity of pseudodifferential operators defined on \(A_ w\), which is an \(L^ 2\)- puzzle. The third section deals with conditions for \(A_ w\) to be an algebra and with those which make an \(A_ w\) an \(L^ 2\)-puzzle when it is an algebra. In the fourth section a sufficient condition is given for a pseudodifferential operator on a generalized Beurling algebra to be continuous. Multipliers are studied in detail for two special classes of examples. In the final section he discusses certain weighted \(L^ 2\)- spaces which appear to be the Fourier transform of the spaces of the form \(A_ w\). An application to BMO is described.

Reviewer: M.Hasumi

##### MSC:

46J10 | Banach algebras of continuous functions, function algebras |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

42B15 | Multipliers for harmonic analysis in several variables |