[PDF] - The Muckenhoupt-type estimations for the best constants in PDF Document

SLIDE 1

The Muckenhoupt-type estimations for the best constants in multidimensional modular inequalities

Chang-Pao Chen Department of Applied Mathematics Hsuan Chuang University Hsinchu, Taiwan 30092, R.O.C. Email: cpchen@wmail.hcu.edu.tw and Jin-Wen Lan Department of Mathematics National Tsing Hua University and Dah-Chin Luor Department of Applied Mathematics I-Shou University

SLIDE 2

Notations

Σn−1: the unit sphere in Rn, E ⊂ Rn: a spherical cone, that is,

each x ∈ E is of the form x = sτ for some 0 < s < ∞ and some τ ∈ A,

A: a given Borel measurable subset of Σn−1, Sx = {y ∈ E : y = sτ, 0 < s < |x|, τ ∈ A}, ˜ Sx = {y ∈ E : y = sτ, 0 < s ≤ |x|, τ ∈ A}, k(x, t) ≥ 0: a locally integrable Borel

measurable function defined on E × E, σ: a σ-finite Borel measure on E.

SLIDE 3

Introduction Consider the integral operator

Kf(x) :=

k(x, t)f(t)dσ(t) (x ∈ E), (1.1) This paper deals with the following modular inequality

q

dµ

1

≤ C

p

dν

1

, (1.3) where Φ ◦ f(x) = Φ(f(x)), 1 ≤ p, q ≤ ∞, µ, ν are two σ-finite Borel measures on E, Φ ∈ CV +(I) for some open interval I in R. Here CV +(I) denotes the class of all nonneg- ative convex functions defined on I. We try to find the smallest constant C such that (1.3) holds for some suitable class of f.

SLIDE 4

The problem that we consider in this paper is to estimate K∗, where K∗ is the “norm” of the operator

K : DK ∩ Lp

DK consists of those f such that Kf(x) is well-defined for µ a.e. x ∈ E, Lp

measurable f with fΦ,p,ν < ∞, fΦ,p,ν :=

p

dν

1/p

< ∞. (1.4)

SLIDE 5

Facts The value of K∗ has been investigated for a long time. (1) It was initiated by the work of Hardy. In [9, Theorem 327], the following inequality was established for 1 < p < ∞:

∞

x

p

dx ≤

p − 1

p ∞

f(x)p dx (f ≥ 0). (1.5) It is known that the constant

p

is the best possible.

SLIDE 6

(2) Hardy’s result was extended in many

(see [3, Theorem 1]) established the following n-dimensional extension of Hardy’s result:

|B(|x|)|

p

dx ≤

p − 1

p Rn f(x)p dx

(f ≥ 0), (1.6) where B(r) is the closed ball centered at the origin with radius r. Remark: This result is the same as to say that K∗ = p/(p − 1), where p = q, Φ(x) = |x|, k(x, t) = 1/|B(|x|)|, dσ = dt, A = Σn−1, E = Rn \ {0}, dµ = dν = dx.

SLIDE 7

(3) Eq. (1.6) has been extended to the case: Φ(x) = |x| of (1.3) for dµ = u(x)dx, dν = v(x)dx, and the so-called Oinarov kernel (cf. [5, 14, 27, 28] for details). A typical result in this direction (see [28, The-

and Φ(x) = |x|, (1.3) holds for all 0 ≤ f ∈ L1

if and only if max(A0, A1) < ∞. Moreover, K∗ ≈ max(A0, A1), where A0 = A0(x)r,ω0, A1 = A1(x)r,ω1, dω0 = v(x)1−p∗dx, dω1 = u(x)dx and A0(x) =

v(t)1−p∗dt

η∗

×

q

u(z)dz

1/q

, (1.7)

SLIDE 8

A1(x) =

p∗

v(t)1−p∗dt

1/p∗

×

u(z)dz

1/η

. (1.8) Here and in sequel, 1/r = 1/q − 1/η, η = max(p, q), and (·)∗ is the conjugate exponent of (·). Remark: For k(x, t) = 1, it is known that A1 = A0 for 1 < p ≤ q < ∞ and A1 = (q/p∗)1/rA0 for 1 < q < p < ∞ (by the integration by parts). In this case, K∗ ≈ A0.

SLIDE 9

(4) Wedestig’s result only says that C1 max(A0, A1) ≤ K∗ ≤ C2 max(A0, A1) for some constants C1 and C2. However, the best possible values of C1 and C2 are still unknown (cf. [14, 28]). In [14, pp.4-5],

ered k(x, t) = 1 and established the following Muckenhoupt-type estimates for the case that n = 1 and Φ(x) = |x|: ρA0 ≤ K∗ ≤

p∗ + q η

1/q

1 + p∗ η

η∗/(p∗q∗)

A0, (1.9) where ρ =

      

1 (p ≤ q),

1/q∗

q1/q (q < p).

SLIDE 10

(5) Later in [8, 27], it was shown that for n ≥ 1, and Φ(x) = |x|, A0 ≤ K∗ ≤ p1/q(p∗)1/p∗A0 for 1 < p ≤ q < ∞ and ρ(q/r)1/qA0 ≤ K∗ ≤ r1/rp1/p(p∗)1/q∗A0 for 1 < q < p < ∞. (6) On the other hand, as indicated in [28, Section 3.4 & Lemma 7.4], there are two other types of estimates instead of the upper bound in (1.9). They are p∗APS and AW := inf

p − s

1/p∗

, (1.10) where 1 < p ≤ q < ∞, APS = sup

v(t)1−p∗dt

−1/p

×

u(t)

v(y)1−p∗dy

q

dt

1/q

,

SLIDE 11

and AW(s) = sup

v(t)1−p∗dt

(s−1)/p

×

u(t)

v(y)1−p∗dy

q(p−s

dt

1/q

. The former was found in Persson-Stepanov [25] and the latter was given in Wedestig [28, The-

Remark: As claimed in [28, pp.27-29], for some case (e.g. p = 3, q = 4), the upper bound estimate given in (1.9) is better than p∗APS. However, up to now, there is no significant result concerning the comparison problem between the right side of (1.9) and (1.10).

SLIDE 12

Goal: The purpose of this paper is to establish the following result for those k satisfying

k(x, t)dσ(t) = 1 (x ∈ E) (∗) Main Result Let 1 ≤ p, q < ∞, Φ ∈ CV +(I), and k(x, t) = g(t)ψ(x, t). Suppose that

dω = 0 for all x ∈ E. (∗∗) Then the following assertions hold: (i) If AM < ∞, then K∗ ≤

p∗ + q η

1/q

1 + p∗ η

η∗/(p∗q∗)

AM. (ii) The condition (∗∗) is not necessary for the case 1 < p ≤ q < ∞. For Φ(x) = |x|, the condition (∗) can be removed.

SLIDE 13

(iii) In (i), if dµ(x) = u(x)dx, dν(x) = v(x)dx, dσ(x) = ξ(x)dx, and ψ(x, t) ≤ ψ(x, s) ≤ Dψ(x, t) for |t| ≤ |s| ≤ |x|, then for Φ(x) = |x|, ρ∗AM ≤ K∗ ≤

p∗+q η

1/q

1+p∗ η

η∗/(p∗q∗)

AM, where u(x) ≥ 0, v(x) ≥ 0, and ξ(x) > 0 on E, and ρ∗ =

      

D−1 (p ≤ q), q1/q

1/q∗

(q < p). Notations: AM = A(x)r,ω, dω(t) =

dνa/dσ

p∗

dν(t), A(x) =

dνa/dσ

×

ψ(·, t)

.

SLIDE 14

Case I: 1 < p = q < ∞, dσ = dµ = dν = dt, ψ(x, t) = 1/G(x), where g(x1) ≤ g(x2) (|x1| ≤ |x2|) and 0 < G(x) :=

g(t)dt < ∞ (x ∈ E). The n-dimensional extensions of Levinson’s modular inequality: Let 1 < p < ∞. Then for Φ ∈ CV +(I) and f : E → ¯ I, we have

G(x)

g(t)f(t)dt

p

dx

1/p

≤ p∗

1/p

. Remark: The result of Christ and Grafakos (see [3, Theorem 1]) corresponds to the case: Φ(x) = |x| and g(t) = 1.

SLIDE 15

Case II: p = q = 1, ψ(x, t) = 1/G(x), dσ = dt, dµ(x) = χ˜

dν(x) =

v(x) |x| +χE\˜

ρ(x) k

(k = 1, 2, · · · ). Generalizations of the Hardy-Knopp-type inequalities: Let 0 < G(x) :=

g(t)dt < ∞ (x ∈ E). Suppose that b ∈ E ∪ {∞} and u : ˜ Sb → [0, ∞) be such that

Sb. Then for Φ ∈ CV +(I) and f : ˜ Sb → ¯ I, we have

Φ

G(x)

g(t)f(t)dt

|x| dx ≤

Φ ◦ f(x)v(x) |x| dx, where v(x) = |x|g(x)

SLIDE 16

Case III: 1 ≤ p ≤ q < ∞, Φ(x) = |x|, g(t) = e−m|t|, ψ(x, t) = em|x|, and dσ = dµ = dν = |x|1−ndx, where m > 0. Extension of Stepanov’s result:

em(|x|−|t|)|t|1−nf(t)dt

|x|1−ndx

1/q

≤

m

1/p∗+1/q

1/p

. Remark: The particular case 1 < p = q < ∞ and n = 1 of the above result reduces to [15, Theorem D83], which was attributed by V. I. Levin and S. B. Steˇ ckin to V. V. Stepanov.

SLIDE 17

Case IV: 1 ≤ q < p < ∞, Φ(x) = |x|, g(t) = e−m|t|, ψ(x, t) = em|x|, and dσ = |x|1−ndx, dµ = u(x)|x|1−ndx, dν = v(x)|x|1−ndx, where m > 0. n-dimensional extension of Heinig’s result:

em(|x|−|t|)|t|1−nf(t)dt

u(x)|x|1−ndx

1/q

≤ q1/q(p∗)1/q∗B∗

1/p

, where B∗ =

v(t)1−p∗|t|1−ndt

1/q∗

×

emq(|s|−|x|)u(s)|s|1−nds

1/q

and d˜ ω = v(x)1−p∗|x|1−ndx.

SLIDE 18

Case V: 1 ≤ p ≤ q < ∞, g(t) = 1, ψ(x, t) = (|x| − |t|)m−1, dσ = dt, dµ = u(x)qdx and dν = v(x)pdx, where m > 0. n-dimensional Riemann-Liouville operator:

(|x| − |t|)m−1f(t)dt

u(x)qdx

1

≤

p∗

1

1 + p∗ q

1

AM

1

, where AM = sup

v(t)

p∗

dt

1

×

q

dz

1

and ρ(z, x) =

  

|z|m−1 (m ≥ 1); (|z| − |x|)m−1 (0 < m < 1). Remark: There exists a similar result for 1 ≤ q < p < ∞.

SLIDE 19

Case VI: 1 ≤ p ≤ q < ∞, g(t) = |t|m−1, ψ(x, t) = (1−|x|/|t|)m−1, dσ = dt, dµ = u(x)qdx and dν = v(x)pdx, where m > 0. Weyl fractional integral operator:

(|t| − |x|)m−1f(t)dt

u(x)qdx

1

≤

p∗

1/q

1 + p∗ q

1/p∗

˜ AM

1

, where ˜ AM = sup

v(t)

p∗

dt

1

×

ρ(z, x)u(z)

q

dz

1

and ˜ ρ(z, t) =

  

1 (m ≥ 1); (1 − |z|/|t|)m−1 (0 < m < 1).

SLIDE 20

Comments (a) Our main results can be regarded as the n-dimensional modular forms of [1, 10]. (b) They are generalizations of the result of Muckenhoupt-Bradley-Maz’ja (cf. [2], [19, §1.3], [20]), [3, Theorem 1], [8, Theorem 2.1], and [27, Theorem 2.1]. (c) There exist some cases for which the up- per bound estimate in the main result is fi- nite, but AW = ∞. Consider the example: n = 1, Φ(x) = |x|, g(t) = [(t + 2) ln(t + 2)]−1/q ψ(x, t) = x−1/q ln−2/q(x/t + 1), dσ(t) = dt, dµ(x) = dx, and dν(t) = t(p∗

Here 1 < p ≤ q < p∗.

SLIDE 21

(d) For some case, our estimate is better than the estimate obtained directly from the proof

80-83 & pp.93-96]). Consider the case: 1 < p < q = 2, k(x, t) = 1, dσ = dt, dµ = u(x)dx, and dν = v(x)dx. According to the proofs given in [14, pp. 80- 83 & pp.93-96]), Wedestig’s result gives K∗ ≤

1/2

A0. (3.25) It is easy to see that the right side of (3.25) is greater than (1 + q/p∗)1/q(1 + p∗/q)1/p∗A0 as p is close to 1 and q = 2. This indicates that even for k(x, t) = 1, there exist p and q such that the upper bound estimate in (3.7) is better than the one derived directly from the proof of Wedestig’s result.

SLIDE 22

(e) Our main result allows us to have many choices for k(x, t) = g(t)ψ(x, t). For the choice g(t) = 1 and ψ(x, t) = k(x, t), the correspond- ing Muckenhoupt estimate AM may be infinite. However, if we make a suitable choice for g(t) and ψ(x, t), the result may be completely dif-

k(x, t) = {x(t + 2) ln(t + 2)}−1/q ln−2/q(x/t+1). For the choice g(t) = 1 and ψ(x, t) = k(x, t), AM = ∞. But, if we choose g(t) = [(t + 2) ln(t + 2)]−1/q and ψ(x, t) = x−1/q ln−2/q(x/t + 1), we get AM < ∞, and so the main result works. This indicates that we can use the choice of g(t) and ψ(x, t) to control the upper bound estimate given in the main result.

SLIDE 23

References [1] Andersen, K. F., Heinig, H. P.: Weighted norm inequalities for certain integral operators. Siam J. Math. Anal. 14, no. 4, 834-844 (1983) [2] Bradley, J. S.: Hardy inequalities with mixed

(1978) [3] Christ, M., Grafakos, L.: Best constants for two nonconvolution inequalities. Proc. Amer.

[4] ˇ Ciˇ zmeˇ sija, A., Peˇ cari´ c, J.: Some new gen- eralisations of inequalities of Hardy and Levin- Cochran-Lee. Bull. Austral. Math. Soc. 63, 105-113 (2001)

SLIDE 24

[5] ˇ Ciˇ zmeˇ sija, A., Peˇ cari´ c, J., Peri´ c, I.: Mixed means and inequalities of Hardy and Levin- Cochran-Lee type for multidimensional balls.

2552 (2000) [6] ˇ Ciˇ zmeˇ sija, A., Peˇ cari´ c, J., Persson, L.-E.: On strengthened Hardy and P´

equalities.

Theory 125, 74-84 (2003) [7] Copson, E.T.: Some integral inequalities. Proc. Roy. Soc. Edinburgh. 75A, 157-164 (1975/76) [8] Dr´ abek, P., Heinig, H. P., Kunfer, A.: Higher dimensional Hardy inequality. Intenat. Ser.

[9] Hardy, G. H., Littlewood, J. E., P´

Inequalities, 2nd edition. Cambridge University Press, Cambridge (1967)

SLIDE 25

[10] Heinig, H. P.: Weighted norm inequalities for certain integral operators II. Proc. Amer.

[11] Heinig, H. P.: Modular inequalities for the Hardy averaging operator. Mathematica Bo- hemica, 124, No. 2-3, 231-244 (1999) [12] Kaijser S., Nikolova L, Persson L.-E. and Wedestig A.: Hardy-type inequalities via con- vexity. Math. Inequal. Appl. 8(3), 403-417 (2005) [13] Kaijser, S., Persson, L.-E., ¨ Oberg, A.: On Carleman and Knopp’s inequalities. J. Approx. Theory 117, 140-151 (2002) [14] Kufner, A., Persson, L.-E.: Weighted in- equalities of Hardy type. World Scientific Pub- lishing Co., Singapore, New Jersey, London, Hong Kong (2003)

SLIDE 26

[15] Levin, V. I., Steˇ ckin, S.B.: Inequalities. Amer. Math. Soc. Transl. (2), 14, 1-29 (1960) [16] Levinson, N.: Generalizations of an in- equality of Hardy. Duke Math. J. 31, 389-394 (1964) [17] Love, E. R.: Generalizations of Hardy’s in- tegral inequality. Proc. Roy. Soc. Edinburgh 100A, 237-262 (1985) [18] Manakov, V.M.: On the best constant in weighted inequalities for Riemann-Liouville

448 (1992) [19] Maz’ja, V. G.: Sobolev spaces. Springer- Verlag, Springer Series in Soviet Mathematics (1985)

SLIDE 27

[20] Muckenhoupt, B.: Hardy’s inequality with

[21] Nassyrova, M., Persson, L.-E., and Stepanov, V.D.: On weighted inequalities with geomet- ric mean operator generated by the Hardy-type integral transform.

Pure Appl. Math., 3, Issue 4, Article 48 (2002) (elec- tronic) [22] Oguntuase, J. A., Persson L.-E., Essel,

Multidimensional Hardy-type inequali- ties with general kernels. J. Math. Anal. Appl. 348, 411-418 (2008) [23] Okpoti, C. A., Persson, L.-E., Sinnamon, G.: An equivalence theorem for some inte- gral conditions with general measures related to Hardy’s inequality.

Anal. Appl. 326 398-413 (2007)

SLIDE 28

[24] Okpoti, C. A., Persson, L.-E., Sinnamon, G.: An equivalence theorem for some inte- gral conditions with general measures related to Hardy’s inequality II. J. Math. Anal. Appl. 337 219-230 (2008) [25] Persson, L.-E., Stepanov, V.D.: Weighted integral inequalities with the geometric mean

(2002) [26] Sinnamon, G.: Weighted Hardy and Opial- type inequalities. J. Math. Anal. Appl. 160, 434-445 (1991) [27] Sinnamon, G.: One-dimensional Hardy- type inequalities in many dimensions. Proc.

SLIDE 29

[28] Wedestig, A.: Weighted Inequalities of Hardy-type and their Limiting inequalities. PhD thesis, Lulea University of Technology, Lulea 2003. [29] Wheeden, R. L., Zygmund, A.: ”Measure and Integral,” Marcel Dekker Inc., New York, 1977. [30] Zygmund, A.: Trigonometric series. Cam- bridge Univ. Press, New York (1959)