Square root of a Schrödinger operator and its L^{p} norms

Ichiro Shigekawa (Kyoto University)

(joint work with Tomohiro Miyokawa)

The following norm equivalence is well-known on the Euclidean space:

∥\sqrt{−△}f∥_{p} ∼ ∥∇f∥_{p}.  (1)

Here, △ denotes the Laplacian, ∥ · ∥_{p} denotes the L^{p} norm (1 < p < ∞) and the notation A ∼ B means that cA ≤ B ≤ CA for some constants c > 0 and C > 0 which is independent of f. This equivalence leads the L^{p}-boundedness of the Riesz transform, which is formally expressed by ∇\sqrt{−△}^{−1}. Moreover the equivalence is extended to Riemannian manifolds (at least compact case).

In this talk, we extend this equivalence to the case of Schrödinger operator △ − V on a Riemannian manifold M. Here △ is the Laplace-Beltrami operator and V is a scalar function. We assume the following. First, the Ricci curvature is bounded from below. Second, the potential function V is bounded from below. By adding a positive constant, we can and do assume that V is uniformly positive. This is just for notational convenience. We further assume that ∇V/ max{V, 1} and △V/ max{V, 1} are bounded. Under these conditions we have the following Theorem 1. For 1 < p < ∞, the following norm equivalence holds

∥\sqrt{V − △}f∥_{p} ∼ ∥∇f∥_{p} + ∥\sqrt{V} f∥_{p}, ∀f ∈ C_{0}^{∞}(M).  (2)

To show the theorem above, the following two properties are fundamental.

• the intertwining property

• the Littlewood-Paley inequality The first one takes the following form.

\sqrt{V} (△ − V) = A\sqrt{V} .  (3)

The operator A satisfying this condition is given by

A = △ + b − \frac{1}{2}∇^{∗}b + \frac{1}{4}|b|^{2} − V  (4)

where b = −∇V/V.

To state the second one, we need to introduce the Littlewood-Paley G-functions. They are defined as follows:

G^{→}f(x) = \{∫_{0}^{∞} t|∂_{t}e^{−t\sqrt{V −△}}f(x)|^{2}dt\}^{1/2},
G^{↑}f(x) = \{∫_{0}^{∞} t|∇e^{−t\sqrt{V −△}}f(x)|^{2}dt\}^{1/2},
G^{V} f(x) = \{∫_{0}^{∞} t|\sqrt{V} e^{−t\sqrt{V −△}}f(x)|^{2}dt\}^{1/2}.

We have the following domination which is called the Littlewood-Paley inequality.

Proposition 2. For 1 < p < ∞, it holds that

∥f∥_{p} ≲ ∥G^{→}f∥_{p}, ≲ ∥f∥_{p}
∥G^{↑}f∥_{p} ≲ ∥f∥_{p},
∥G^{V} f∥_{p} ≲ ∥f∥_{p}.

Here the notation A ≲ B means that A ≤ CB for some constants C > 0 which is independent of f.

To combine this with the intertwining property, we need to introduce the Littlewood-Paley G-functions for the operator A. To do this, we just replace △ − V with A and denote the Littlewood-Paley G-functions by G\overrightarrow{A}, G_{A}^{↑}, etc. Similar inequality holds for A, e.g., ∥f∥_{p} ≲ ∥G\overrightarrow{A}f∥_{p} ≲ ∥f∥_{p}. The intertwining property yields that G\overrightarrow{A}\sqrt{V} f = G^{V} f. Using this relation, we can show that

∥\sqrt{V} f∥_{p} ≲ ∥\sqrt{V − △}f∥_{p}.

Remaining inequality can be shown similarly.

So far, we have considered \sqrt{V − △}. If we consider △ − V itself, then we have Theorem 3. For 1 < p < ∞, the following norm equivalence holds

∥(△ − V)f∥_{p} ∼ ∥△f∥_{p} + ∥V f∥_{p}, ∀f ∈ C_{0}^{∞}(M).  (5)

We can also extend the above theorem for the Hodge-Kodaira operator dd^{∗} + d^{∗}d plus the potential V. In this case, we need the positivity of the Riemannian curvature.

References

[1] D. Bakry, Transformations de Riesz pour les semigroupes symétriques, Seconde partie: etude sous la condition Γ_{2} ≥ 0, Séminaire de Prob. XIX, pp. 145–174, Lecture Notes in Math., vol. 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985.

[2] I. Shigekawa, L^{p} contraction semigroups for vector valued functions, J. Funct. Anal., 147 (1997), 69–108.

[3] I. Shigekawa, Littlewood-Paley inequality for a diffusion satisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary, to appear in Osaka J. Math.

[4] E. M. Stein, “Topics in harmonic analysis, related to Littlewood-Paley theory,” Annals of Math. Studies, 63, Princeton Univ. Press, 1974.

[5] N. Yoshida, The Littlewood-Paley-Stein inequality on an infinite dimensional manifold, J. Funct. Anal., 122 (1994), 402–427.