Construction of Weights for Positive Integral Operators

: Let ( X , M , µ ) be a σ -ﬁnite measure space and denote by P ( X ) the µ -measurable functions f : X → [ 0, ∞ ] , f < ∞ µ ae. Suppose K : X × X → [ 0, ∞ ) is µ × µ -measurable and deﬁne the mutually transposed operators T and T (cid:48) on P ( X ) by ( T f )( x ) = (cid:82) X K ( x , y ) f ( y ) d µ ( y ) and ( T (cid:48) g )( y ) = (cid:82) X K ( x , y ) g ( x ) d µ ( x ) , f , g ∈ P ( X ) , x , y ∈ X . Our interest is in inequalities involving a ﬁxed (weight) function w ∈ P ( X ) and an index p ∈ ( 1, ∞ ) such that: (*): (cid:82) X [ w ( x )( T f )( x )] p d µ ( x ) (cid:46) C (cid:82) X [ w ( y ) f ( y )] p d µ ( y ) . The constant C > 1 is to be independent of f ∈ P ( X ) . We wish to construct all w for which (*) holds. T φ 1 ≤ C 1 φ 1 and T (cid:48) φ 2 ≤ C 2 φ 2 . Our fundamental result shows that the φ 1 and φ 2 above are within constant multiples of (**): ψ 1 + ∑ ∞ j = 1 E − j T ( j ) ψ 1 and ψ 2 + ∑ ∞ j = 1 E − j T (cid:48) ( j ) ψ 2 respectively; here ψ 1 , ψ 2 ∈ P ( X ) , E > 1 and T ( j ) , T (cid:48) ( j ) are the j th iterates of T and T (cid:48) . This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are deﬁned through symmetric kernels K ( x , y ) = K ( y , x ) , so that T (cid:48) = T . This means that only the ﬁrst series in (**) needs to be studied.


Further, (1) holds if and only if the dual inequality
does.
In this paper, we are interested in constructing weights u and v for which (1) holds. We restrict attention the case u = v = w; the general case will be investigated in the future. Our approach is based on the observation that, implicit in a proof of the converse of Schur's lemma, given in [10], is a method for constructing w. An interesting application of Schur's lemma itself to weighted norm inequalities is given in Christ [11].
In Section 2, we prove a number of general results the first of which is the following one.
Theorem 1. Let (X, M, µ) be a σ-finite measure space with u, v ∈ P(X), 0 ≤ u, v < ∞, µ ae. Suppose that T is a positive integral operator on P(X) with transpose T . Then, for fixed p, 1 < p < ∞, one has (1), with C > 1 independent of f ∈ P(X), if and only if them exists a function φ ∈ P(X) and a constant C > 1 for which In this case, B 0 , the smallest B possible in (1) and Co, the smallest possible C so that (3) holds for some φ, satisfy for some C > 1.
Though it is often possible to work with the inequalities (4) directly (see Remark 1) it is important to have a general method to construct the functions φ 1 and φ 2 . This method is given in our principal result. Theorem 2. Suppose X, µ and T are as in Theorem 1. Let φ∈P(X). Then, φ satisfies an inequality of the form if and only if there is a constant C > 1 such that where ψ ∈ P(X), C 2 > 1 is constant and T (j) = T • T · · · • T, j times.
The kernels of operator of the form will be called the weight generating kernels of T. In Sections 3-6 these kernels will be calculated for particular T. All but the Hardy operators considered in Section 6 operate on the class P(R n ) of nonnegative, Lebesgue-measurable functions on R n . The operators last referred to are, in fact, convolution operators with even integrable kernels k, R n k(y) dy = 1. In particular, the kernel k(x − y) is symmetric, so T k = T k , whence only the first series in (**) need be considered. Further, the convolution kernels are part of an approximate identity {k t } t>0 on L P (R n ) = f Leb. meas: see [12]. Thus, it becomes of interest to characterize the weights w for which {k t } l>0 is an approximate identity on that is k t * f ∈ L p (w) and lim It is a consequence of the Banach-Steinhaus Theorem that this will be so if and only if sup 0<t<a k t < ∞ for some fixed a > 0, where k t denotes the operator norm of T k t on L p (w). We remark here that the operators in Sections 3-5 are bounded on L p (w) and, indeed, form part of an approximate identity on L p (w), if w satisfies the A p condition, namely, the supremum being taken over all cubes Q in R n whose sides are parallel to the coordinate axes with ∞ > |Q| = Lebesgue measure of Q. See ( [13], p. 62) and [14]. Finally, all the convolution operators are part of a convolution semigroup (k t ) t>0 ; that is k t (x) = t −n k x t and k t 1 * k t 2 = k t 1 +t 2 , t 1 , t 2 > 0. The approximate identity result can thus be interpreted as the continuity of the semigroup.
We conclude the introduction with some remarks on terminology and notation. The fact that T is bounded on L p (w) if and only if T is bounded on L p (w −1 ) is called the principle of duality or, simply, duality. Two functions f , g ∈ P(X) are said to be equivalent if a constant C > 1 exists for which We indicate this by f ≈ g, with the understanding that C is independent of all parameters appearing, (except dimension) unless otherwise stated. If only one of the inequalities in (8) holds, we use the notation f g or f g, as appropriate. Lastly, a convolution operator and its kernel are frequently denoted by the same symbol.

General Results
In this section we give the proofs of the results stated in the Introduction, together with some remarks.
It will suffice to deal with the first condition in (3). So, Fubini's Theorem yields and hence to since f is arbitrary.
According to the main result of [15], then, i.e., T : L p (v) → L p (u), with norm ≤ C, so that (1) holds with B ≤ C.

Remark 1. The class of functions φ determined by the weight-generating operators
This means that in dealing with weight-generating operators we need only consider C > 1.
We conclude this section with the following observations on approximate identities in weighted Lebesgue spaces.

Remark 2.
Suppose {k t } t>0 is an approximate identity in L p (R n ), 1 < p < ∞. If the inequalities (4) involving φ 1 and φ 2 can be shown to hold for T kt , t ∈ (0, a] for some a > 0, with C > 1 independent of such t, then {k t } t>0 will also be an approximate identity in L p (w) = L p (R n , w), Example 1. Let k = k(|x|) be any bounded, nonnegative radial function on R n which is a decreasing function of |x| and suppose To obtain the boundedness assertion take φ 1 (x) = 1 and φ 2 (x) = 1 + |x| −n (1 + log + (1/|x|)) −p in Corollary 1.
Arguments similar to those in [6] show that if {k t } t>0 is an approximate identity in L p (w), then w must satisfy the A p condition for all cubes Q will sides parallel to the coordinate axes and |Q| ≤ a for some a > 0. However, the weight w does not have this property.

The Poisson Integral Operators
We recall that for t > 0 and y ∈ R n , the Poisson kernel, P t , is defined by Theorem 3. The weight-generating kernels for P t , t > 0, are equivalent to P ≡ P 0 . Indeed, given ψ ∈ P(R n ), with P ψ < ∞ a.e., where C > 1, Proof. Observe that by the semigroup property P (j) As stated in Section 1, w ∈ A p is sufficient for {P t } t>0 to be an approximate identify in L P (w). Moreover, w ∈ A p is also necessary for this in the periodic case. See [6,8,16]. It is perhaps surprising then that the class of approximate identity weights is much larger than A p , as is seen in the next result.
for all f ∈ L p (ω α ). The set of α for which w α ∈ A p , however, is Proof. We omit the easy proof of the assertion concerning the α for which w α ∈ A p . To obtain the "if" part of the other assertion we will show if and only if −n − 1 ≤ β < 1, with C > 1 independent of both s and t, if t ∈ (0, 1). Corollary 1 and Remark 2, then yield (10) when − n p − 1 < α < n p + 1. Consider, then, fixed x ∈ R n and 0 < t < 1. We have Now, Again, (11) is to hold. Moreover, for x ∈ R n and 0 < t < 1, Next, for |x| 1 which requires β < 1 to have I 3 < ∞. In that case That P t is not bounded on L p (w α ) when α ≤ − n p − 1 can be seen by noting that, for appropriate ε > 0, the function f (x) = |x|[log(1 + |x|)] −(1+ε)/p is in L p (w α ), while P t f ≡ ∞. The range α ≥ n/p + 1 is then ruled out by duality.

The Bessel Potential Operators
The Bessel kernel, G α , α > 0, can be defined explicitly by G α (y) = C α |y| (α−n)/2 K (n−α)/2 (|y|), y ∈ R n , where K r is the modified Bessel function of the third kind and It is, however, more readily recognized by its Fourier transformation Using the latter formula one picks out the special cases G n−1 and G n+1 which, except for constant multiplies, are, respectively, |y| −1 e −|y| and the Picard kernel e −|y| .
The semigroup properly G α * G β = G α+β holds and so the jth convolution iterate has kernel G jα .
We use the integral representation to show in Lemma 1 below that known estimates [17], are in fact, sharp.
We seek conditions on r and δ so that if δ − αp + γ − n ≥ 0. By taking γ sufficiently close to β and δ sufficiently closed to n, this condition can be met. In particular, for α ∈ (0, 2], the kernel is equivalent to G α (ky) + G 2 (ky).
Thus, the kernel is, essentially, Now, the first term in (17) is bounded on 0 ≤ r ≤ 1, while the second term is equivalent to G α for all r ≥ 0. It only remains to show the first integral, I, satisfies I ≈ r (1−n)/2 e −kr for r ≥ 1. To this end set s = rt/2 in I to obtain where β(r) = √ cr/2 − k √ 2/cr and f (y) = y 2 + 4l = √ s + k √ s . Finally, take z = √ r/2y to get We have now just to observe that when z ∈ R and r ≥ 1 Typical of G α weights are the exponential functions e βx , −1 < β < 1.

Example 2.
Consider the Bessel potential G 2 (y) so that the weight-generating kernels are equivalent to G 2 (ky), 0 < k < 1. These are especially simple when the dimension, n, is 1 or 3. In the first case G 2 (y) is essentially equal to the Picard kernel, e −|y| , and in the second case to |y| −1 e −|y| . According to Corollary 1, then, T G 2 is bounded on L p (e k/p |y| ) and L p (e −k/p|y| ) when n = 1; on L p |y| 1/p e k/p |y| and L p |y| 1/p e −k/p|y| when n = 3.
Theorem 5. Fix C > 1. Then, the weight-generating kernel for W l corresponding to C is equivalent to with the constants of equivalence independent of t ∈ (0, a), |y| > 4ka 1/2 , where 0 < a < 1.

Remark 4.
The weight-generating kernels are similar to those of G 2 on R 1 and R 3 (see Example 2), whence the exponential weights of Proposition 4 are in some sense typical. This illustrates a general theorem of Lofstrom, [18], which asserts that no translation-invariant operator is bounded on L p (w), when w is a rapidly varying weight such as w(α) = exp(|x| α ), α > 1.
Finally, we denote the product x −1 1 . . . x −1 n by x −1 or 1 x and the product (log x 1 y 1 ) . . . (log x n y n ) by log x y ; here, x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) belong to R n + . The Hardy averaging operators, P n and Q n , are defined at f ∈ P(R n + ), x ∈ R n + , by These operators, which are the transposes of one another, are generalizations to n-dimensions of the well-known ones, considered in [5] for example. A simple induction argument leads to the following formulas for the iterates of P n and Q n : in which x ∈ R n + and j = 0, 1, . . . . From Theorem 1 of [19], we obtain the representations of the weight-generating kernels of P n and Q n described below. Theorem 6. For C > 1 and set α = nC −1/n . Then, the weight-generating kernels for P n and Q n corresponding to C are equivalent, respectively, to and Then P n is bounded on L p (w β ) if and only if β < 1/p ; by duality, Q n is bounded on L p (w β ) of and only if β > −1/p.
Moreover, any weight w satisfying (22) is equivalent to one in the above form.
Proof. This result is a consequence of Corollary 1 and Theorem 2.
Added in Proof: While this work was in press the author came across the paper [20]. In it Bloom proves our Theorem 1 using complex interpolation rather than interpolation with change of measure. A (typical) application of his result to the Hardy operators substitutes them in the necessary and sufficient conditions, thereby giving a criterion for their two weighted boundedness. This is in contrast to our Theorem 6, in which the explicit form of a single weight is given.
Funding: This research received no external funding.