estimates for the 9-neumann operator ln weighted hilbert spaces

TRANSACTIONS OF THE AMERICAN MATHEMATICALSOCIETY Volume 217, 1976

ESTIMATESFOR THE 9-NEUMANNOPERATOR LNWEIGHTEDHILBERTSPACES BY SIDNEY L. HANTLER ABSTRACT.

Estimates for the Ü operator are used to derive estimates for

the Neumann operator in weighted Hilbert spaces. The technique is similar to

that used to prove regularity of solutions of elliptic partial differential equations.

A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions.

1. Introduction. L. Hörmander gave fundamental estimates for the 9 operator in weighted Hilbert spaces in [5]. Based on this work, we give here improved estimates for the associated Neumann operator. These estimates are applied to give new bounds for the reproducing kernels of the subspaces of analytic functions. In [3], this technique is applied to problems of weighted polynomial approximation. To introduce our results, we present some notation. The Hilbert space of (classes of) functions on C" which are square integrable with respect to the measure e-0} dX, where co is measurable with respect to the Lebesgue measure dX, is denoted by Z,2(co). Similarly L2 „)()is the space of differential forms of type {p, q) whose coefficients belong to Z,2(co). To illustrate our results, suppose n = 1 and there is a positive constant c so that 92co/9z9z > c. Then the Neumann operator, N, maps Z,?0 i)(co) into Z,20jx(co). It is shown, in Theorem 2, that if

aff+1)(02)into L2pq+1)(#2)- The operator N is the Neumann operator on Lfaq+D^) and \\N\\ < 2/c. Our object is to prove that the Neumann opera-

tor maps certain proper subspaces oíL2pq+1-){(p2)into other proper subspacesof

¿(p.í+dO^)3. A priori estimates in a special case. In this section we obtain a priori estimates for the Neumann operator when N{0) has smooth, compactly supported coefficients. Strong a priori estimates cannot be extended directly due to difficulties encountered in proving suitable approximation results. In §4 an approximation result is proved which enables us to prove weak estimates for a more general class of forms. These estimates are then used to prove stronger estimates for this larger class of forms. Certain technical difficulties arise in proving estimates for the Neumann operator when n > 1 and i// =£0. These details obscure the general argument and, therefore, in this section and in §4 we assume that « = 1 and \¡/= 0. We also assume that p = a = 0. In §5, we indicate how these restrictions can be removed. General results with references to complete proofs are presented in §6.

Recallthat 0 G C2{C)and that 920/9z9z > c > 0 from (2.1). Because n = 1, we have 5 = 0 and, thus, N is the inverse of TT*. Where there is no

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398

S. L. HANTLER

danger of confusion we write / = fdz.

(3•!)

It follows from (4.1.9) in [4] that

T*if) = (90/3z)/- df/bz whenfEDT*

and thus that

(3.2)

TT\f)

= (92ç4/3z9z)/+ T*idf/dz)dz

when/£ DTT*.

We conclude that

(3.3)

T*iGf) = GT*if) - (9G/9z)/ when/ and Gf are in DTT*

and that

^^

7TW) = GTT*if)+ § T*if)dz - g/-

g §¿z

when / and G/ are in Drr*.

Finally, observe that if fEDTT*, we have

(3-5)

ll/ll0< IIAWr*(/)||0

and

(3-6)

lir*(/)ii0 < ii7vii,/2iirr*(/)ii0.

We can now obtain a priori estimates for the Neumann operator by considering the commutator of TT* and multiplication by a smooth function. Let

6 E L\0A)(1) and G £ C°°iC). Notice that

(3.7)

GW(0)= NiGO)+ N[TT*, G]A/(0)

where [A, B] denotes the commutator AB - BA. This yields

||GAf(0)||0< ||A>||||G0||0+ \\N\\\\[TT*,G]Af(0)||0. Our goal is to choose G and a subspace H of ¿2o>i)(0) containing P(o,i) so that the right-hand side of (3.7) is finite when 0 £ H. We then prove an approximation result on H and apply a weak compactness argument to conclude that GN(0)

G¿(o,i)(0)As an example, suppose that A^(0)G í\o,i)> that 0 0 in L2QtXß>)for / = 0, 1, . . . , /. By (3.11), we have

\\TT*izlfp)-TT*iz'fq)\\4) l

IIAfll-//2

/=o

I-

1)(c6 - ß\z\). Then Nid) E ¿2O(,)(0 - a|z|) and T*Nid) EL2i- a\z\).


402

S. L. HANTLER

Proof.

Notice that for all nonnegative integers k, |z|fce~^|z|/2 /2||0||0_ß|zl

The proof is completed by observing that these inequalities insure that the power series for ea ^z}^N{d) convergesin L201^{1)then N{0) G

¿(o.i/2*2 - lzDand T*N(ß)E ¿2(2*2 - l*D5. The general method. In § §3 and 4 we used an a priori estimate and an approximation result to obtain estimates for the Neumann operator when n = 1, \b = 0 and p = a = 0. The fact that \p - 0 governed the choice of c7(z) = z in the a priori estimate (3.8). In this section we derive a generalization of (3.8) and discuss the associated approximation result needed to prove estimates in the absence of these restrictions.

Supposenow that 0 G L\pq+lp2),

N{9) E Vlp>q+ l), and G E C°{Cn).

A natural generalization of (3.8) is

\\GNi9)\\^ < ||/V||||GÖ||02+ IUv-|lll[7T* + 5*5, G]/V(0)||02. Unfortunately, we cannot derive the estimates for the commutator [7T* + S*S, G] required to obtain a useful a priori estimate.


^NEUMANN OPERATOR IN WEIGHTEDHILBERT SPACES

403

Observethat (2.1) insures that the operator SS* + L*L has a bounded inverse, M, where L is the weak extension of the 9 operator defined on ^(p,o+2)(^3) "^ ^(p,